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                    GNU GENERAL PUBLIC LICENSE
                       Version 3, 29 June 2007
 Copyright (C) 2007 Free Software Foundation, Inc. <https://fsf.org/>
 Everyone is permitted to copy and distribute verbatim copies
 of this license document, but changing it is not allowed.
                            Preamble
  The GNU General Public License is a free, copyleft license for
 software and other kinds of works.
  The licenses for most software and other practical works are designed
 to take away your freedom to share and change the works.  By contrast,
 the GNU General Public License is intended to guarantee your freedom to
 share and change all versions of a program--to make sure it remains free
 software for all its users.  We, the Free Software Foundation, use the
 GNU General Public License for most of our software; it applies also to
 any other work released this way by its authors.  You can apply it to
 your programs, too.
  When we speak of free software, we are referring to freedom, not
 price.  Our General Public Licenses are designed to make sure that you
 have the freedom to distribute copies of free software (and charge for
 them if you wish), that you receive source code or can get it if you
 want it, that you can change the software or use pieces of it in new
 free programs, and that you know you can do these things.
  To protect your rights, we need to prevent others from denying you
 these rights or asking you to surrender the rights.  Therefore, you have
 certain responsibilities if you distribute copies of the software, or if
 you modify it: responsibilities to respect the freedom of others.
  For example, if you distribute copies of such a program, whether
 gratis or for a fee, you must pass on to the recipients the same
 freedoms that you received.  You must make sure that they, too, receive
 or can get the source code.  And you must show them these terms so they
 know their rights.
  Developers that use the GNU GPL protect your rights with two steps:
 (1) assert copyright on the software, and (2) offer you this License
 giving you legal permission to copy, distribute and/or modify it.
  For the developers' and authors' protection, the GPL clearly explains
 that there is no warranty for this free software.  For both users' and
 authors' sake, the GPL requires that modified versions be marked as
 changed, so that their problems will not be attributed erroneously to
 authors of previous versions.
  Some devices are designed to deny users access to install or run
 modified versions of the software inside them, although the manufacturer
 can do so.  This is fundamentally incompatible with the aim of
 protecting users' freedom to change the software.  The systematic
 pattern of such abuse occurs in the area of products for individuals to
 use, which is precisely where it is most unacceptable.  Therefore, we
 have designed this version of the GPL to prohibit the practice for those
 products.  If such problems arise substantially in other domains, we
 stand ready to extend this provision to those domains in future versions
 of the GPL, as needed to protect the freedom of users.
  Finally, every program is threatened constantly by software patents.
 States should not allow patents to restrict development and use of
 software on general-purpose computers, but in those that do, we wish to
 avoid the special danger that patents applied to a free program could
 make it effectively proprietary.  To prevent this, the GPL assures that
 patents cannot be used to render the program non-free.
  The precise terms and conditions for copying, distribution and
 modification follow.
                       TERMS AND CONDITIONS
  0. Definitions.
  "This License" refers to version 3 of the GNU General Public License.
  "Copyright" also means copyright-like laws that apply to other kinds of
 works, such as semiconductor masks.
  "The Program" refers to any copyrightable work licensed under this
 License.  Each licensee is addressed as "you".  "Licensees" and
 "recipients" may be individuals or organizations.
  To "modify" a work means to copy from or adapt all or part of the work
 in a fashion requiring copyright permission, other than the making of an
 exact copy.  The resulting work is called a "modified version" of the
 earlier work or a work "based on" the earlier work.
  A "covered work" means either the unmodified Program or a work based
 on the Program.
  To "propagate" a work means to do anything with it that, without
 permission, would make you directly or secondarily liable for
 infringement under applicable copyright law, except executing it on a
 computer or modifying a private copy.  Propagation includes copying,
 distribution (with or without modification), making available to the
 public, and in some countries other activities as well.
  To "convey" a work means any kind of propagation that enables other
 parties to make or receive copies.  Mere interaction with a user through
 a computer network, with no transfer of a copy, is not conveying.
  An interactive user interface displays "Appropriate Legal Notices"
 to the extent that it includes a convenient and prominently visible
 feature that (1) displays an appropriate copyright notice, and (2)
 tells the user that there is no warranty for the work (except to the
 extent that warranties are provided), that licensees may convey the
 work under this License, and how to view a copy of this License.  If
 the interface presents a list of user commands or options, such as a
 menu, a prominent item in the list meets this criterion.
  1. Source Code.
  The "source code" for a work means the preferred form of the work
 for making modifications to it.  "Object code" means any non-source
 form of a work.
  A "Standard Interface" means an interface that either is an official
 standard defined by a recognized standards body, or, in the case of
 interfaces specified for a particular programming language, one that
 is widely used among developers working in that language.
  The "System Libraries" of an executable work include anything, other
 than the work as a whole, that (a) is included in the normal form of
 packaging a Major Component, but which is not part of that Major
 Component, and (b) serves only to enable use of the work with that
 Major Component, or to implement a Standard Interface for which an
 implementation is available to the public in source code form.  A
 "Major Component", in this context, means a major essential component
 (kernel, window system, and so on) of the specific operating system
 (if any) on which the executable work runs, or a compiler used to
 produce the work, or an object code interpreter used to run it.
  The "Corresponding Source" for a work in object code form means all
 the source code needed to generate, install, and (for an executable
 work) run the object code and to modify the work, including scripts to
 control those activities.  However, it does not include the work's
 System Libraries, or general-purpose tools or generally available free
 programs which are used unmodified in performing those activities but
 which are not part of the work.  For example, Corresponding Source
 includes interface definition files associated with source files for
 the work, and the source code for shared libraries and dynamically
 linked subprograms that the work is specifically designed to require,
 such as by intimate data communication or control flow between those
 subprograms and other parts of the work.
  The Corresponding Source need not include anything that users
 can regenerate automatically from other parts of the Corresponding
 Source.
  The Corresponding Source for a work in source code form is that
 same work.
  2. Basic Permissions.
  All rights granted under this License are granted for the term of
 copyright on the Program, and are irrevocable provided the stated
 conditions are met.  This License explicitly affirms your unlimited
 permission to run the unmodified Program.  The output from running a
 covered work is covered by this License only if the output, given its
 content, constitutes a covered work.  This License acknowledges your
 rights of fair use or other equivalent, as provided by copyright law.
  You may make, run and propagate covered works that you do not
 convey, without conditions so long as your license otherwise remains
 in force.  You may convey covered works to others for the sole purpose
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 the terms of this License in conveying all material for which you do
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 for you must do so exclusively on your behalf, under your direction
 and control, on terms that prohibit them from making any copies of
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  Conveying under any other circumstances is permitted solely under
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  3. Protecting Users' Legal Rights From Anti-Circumvention Law.
  No covered work shall be deemed part of an effective technological
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 similar laws prohibiting or restricting circumvention of such
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  When you convey a covered work, you waive any legal power to forbid
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 modification of the work as a means of enforcing, against the work's
 users, your or third parties' legal rights to forbid circumvention of
 technological measures.
  4. Conveying Verbatim Copies.
  You may convey verbatim copies of the Program's source code as you
 receive it, in any medium, provided that you conspicuously and
 appropriately publish on each copy an appropriate copyright notice;
 keep intact all notices stating that this License and any
 non-permissive terms added in accord with section 7 apply to the code;
 keep intact all notices of the absence of any warranty; and give all
 recipients a copy of this License along with the Program.
  You may charge any price or no price for each copy that you convey,
 and you may offer support or warranty protection for a fee.
  5. Conveying Modified Source Versions.
  You may convey a work based on the Program, or the modifications to
 produce it from the Program, in the form of source code under the
 terms of section 4, provided that you also meet all of these conditions:
    a) The work must carry prominent notices stating that you modified
    it, and giving a relevant date.
    b) The work must carry prominent notices stating that it is
    released under this License and any conditions added under section
    7.  This requirement modifies the requirement in section 4 to
    "keep intact all notices".
    c) You must license the entire work, as a whole, under this
    License to anyone who comes into possession of a copy.  This
    License will therefore apply, along with any applicable section 7
    additional terms, to the whole of the work, and all its parts,
    regardless of how they are packaged.  This License gives no
    permission to license the work in any other way, but it does not
    invalidate such permission if you have separately received it.
    d) If the work has interactive user interfaces, each must display
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  A compilation of a covered work with other separate and independent
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 used to limit the access or legal rights of the compilation's users
 beyond what the individual works permit.  Inclusion of a covered work
 in an aggregate does not cause this License to apply to the other
 parts of the aggregate.
  6. Conveying Non-Source Forms.
  You may convey a covered work in object code form under the terms
 of sections 4 and 5, provided that you also convey the
 machine-readable Corresponding Source under the terms of this License,
 in one of these ways:
    a) Convey the object code in, or embodied in, a physical product
    (including a physical distribution medium), accompanied by the
    Corresponding Source fixed on a durable physical medium
    customarily used for software interchange.
    b) Convey the object code in, or embodied in, a physical product
    (including a physical distribution medium), accompanied by a
    written offer, valid for at least three years and valid for as
    long as you offer spare parts or customer support for that product
    model, to give anyone who possesses the object code either (1) a
    copy of the Corresponding Source for all the software in the
    product that is covered by this License, on a durable physical
    medium customarily used for software interchange, for a price no
    more than your reasonable cost of physically performing this
    conveying of source, or (2) access to copy the
    Corresponding Source from a network server at no charge.
    c) Convey individual copies of the object code with a copy of the
    written offer to provide the Corresponding Source.  This
    alternative is allowed only occasionally and noncommercially, and
    only if you received the object code with such an offer, in accord
    with subsection 6b.
    d) Convey the object code by offering access from a designated
    place (gratis or for a charge), and offer equivalent access to the
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    e) Convey the object code using peer-to-peer transmission, provided
    you inform other peers where the object code and Corresponding
    Source of the work are being offered to the general public at no
    charge under subsection 6d.
  A separable portion of the object code, whose source code is excluded
 from the Corresponding Source as a System Library, need not be
 included in conveying the object code work.
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 tangible personal property which is normally used for personal, family,
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 modification has been made.
  If you convey an object code work under this section in, or with, or
 specifically for use in, a User Product, and the conveying occurs as
 part of a transaction in which the right of possession and use of the
 User Product is transferred to the recipient in perpetuity or for a
 fixed term (regardless of how the transaction is characterized), the
 Corresponding Source conveyed under this section must be accompanied
 by the Installation Information.  But this requirement does not apply
 if neither you nor any third party retains the ability to install
 modified object code on the User Product (for example, the work has
 been installed in ROM).
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 requirement to continue to provide support service, warranty, or updates
 for a work that has been modified or installed by the recipient, or for
 the User Product in which it has been modified or installed.  Access to a
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 adversely affects the operation of the network or violates the rules and
 protocols for communication across the network.
  Corresponding Source conveyed, and Installation Information provided,
 in accord with this section must be in a format that is publicly
 documented (and with an implementation available to the public in
 source code form), and must require no special password or key for
 unpacking, reading or copying.
  7. Additional Terms.
  "Additional permissions" are terms that supplement the terms of this
 License by making exceptions from one or more of its conditions.
 Additional permissions that are applicable to the entire Program shall
 be treated as though they were included in this License, to the extent
 that they are valid under applicable law.  If additional permissions
 apply only to part of the Program, that part may be used separately
 under those permissions, but the entire Program remains governed by
 this License without regard to the additional permissions.
  When you convey a copy of a covered work, you may at your option
 remove any additional permissions from that copy, or from any part of
 it.  (Additional permissions may be written to require their own
 removal in certain cases when you modify the work.)  You may place
 additional permissions on material, added by you to a covered work,
 for which you have or can give appropriate copyright permission.
  Notwithstanding any other provision of this License, for material you
 add to a covered work, you may (if authorized by the copyright holders of
 that material) supplement the terms of this License with terms:
    a) Disclaiming warranty or limiting liability differently from the
    terms of sections 15 and 16 of this License; or
    b) Requiring preservation of specified reasonable legal notices or
    author attributions in that material or in the Appropriate Legal
    Notices displayed by works containing it; or
    c) Prohibiting misrepresentation of the origin of that material, or
    requiring that modified versions of such material be marked in
    reasonable ways as different from the original version; or
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    those licensors and authors.
  All other non-permissive additional terms are considered "further
 restrictions" within the meaning of section 10.  If the Program as you
 received it, or any part of it, contains a notice stating that it is
 governed by this License along with a term that is a further
 restriction, you may remove that term.  If a license document contains
 a further restriction but permits relicensing or conveying under this
 License, you may add to a covered work material governed by the terms
 of that license document, provided that the further restriction does
 not survive such relicensing or conveying.
  If you add terms to a covered work in accord with this section, you
 must place, in the relevant source files, a statement of the
 additional terms that apply to those files, or a notice indicating
 where to find the applicable terms.
  Additional terms, permissive or non-permissive, may be stated in the
 form of a separately written license, or stated as exceptions;
 the above requirements apply either way.
  8. Termination.
  You may not propagate or modify a covered work except as expressly
 provided under this License.  Any attempt otherwise to propagate or
 modify it is void, and will automatically terminate your rights under
 this License (including any patent licenses granted under the third
 paragraph of section 11).
  However, if you cease all violation of this License, then your
 license from a particular copyright holder is reinstated (a)
 provisionally, unless and until the copyright holder explicitly and
 finally terminates your license, and (b) permanently, if the copyright
 holder fails to notify you of the violation by some reasonable means
 prior to 60 days after the cessation.
  Moreover, your license from a particular copyright holder is
 reinstated permanently if the copyright holder notifies you of the
 violation by some reasonable means, this is the first time you have
 received notice of violation of this License (for any work) from that
 copyright holder, and you cure the violation prior to 30 days after
 your receipt of the notice.
  Termination of your rights under this section does not terminate the
 licenses of parties who have received copies or rights from you under
 this License.  If your rights have been terminated and not permanently
 reinstated, you do not qualify to receive new licenses for the same
 material under section 10.
  9. Acceptance Not Required for Having Copies.
  You are not required to accept this License in order to receive or
 run a copy of the Program.  Ancillary propagation of a covered work
 occurring solely as a consequence of using peer-to-peer transmission
 to receive a copy likewise does not require acceptance.  However,
 nothing other than this License grants you permission to propagate or
 modify any covered work.  These actions infringe copyright if you do
 not accept this License.  Therefore, by modifying or propagating a
 covered work, you indicate your acceptance of this License to do so.
  10. Automatic Licensing of Downstream Recipients.
  Each time you convey a covered work, the recipient automatically
 receives a license from the original licensors, to run, modify and
 propagate that work, subject to this License.  You are not responsible
 for enforcing compliance by third parties with this License.
  An "entity transaction" is a transaction transferring control of an
 organization, or substantially all assets of one, or subdividing an
 organization, or merging organizations.  If propagation of a covered
 work results from an entity transaction, each party to that
 transaction who receives a copy of the work also receives whatever
 licenses to the work the party's predecessor in interest had or could
 give under the previous paragraph, plus a right to possession of the
 Corresponding Source of the work from the predecessor in interest, if
 the predecessor has it or can get it with reasonable efforts.
  You may not impose any further restrictions on the exercise of the
 rights granted or affirmed under this License.  For example, you may
 not impose a license fee, royalty, or other charge for exercise of
 rights granted under this License, and you may not initiate litigation
 (including a cross-claim or counterclaim in a lawsuit) alleging that
 any patent claim is infringed by making, using, selling, offering for
 sale, or importing the Program or any portion of it.
  11. Patents.
  A "contributor" is a copyright holder who authorizes use under this
 License of the Program or a work on which the Program is based.  The
 work thus licensed is called the contributor's "contributor version".
  A contributor's "essential patent claims" are all patent claims
 owned or controlled by the contributor, whether already acquired or
 hereafter acquired, that would be infringed by some manner, permitted
 by this License, of making, using, or selling its contributor version,
 but do not include claims that would be infringed only as a
 consequence of further modification of the contributor version.  For
 purposes of this definition, "control" includes the right to grant
 patent sublicenses in a manner consistent with the requirements of
 this License.
  Each contributor grants you a non-exclusive, worldwide, royalty-free
 patent license under the contributor's essential patent claims, to
 make, use, sell, offer for sale, import and otherwise run, modify and
 propagate the contents of its contributor version.
  In the following three paragraphs, a "patent license" is any express
 agreement or commitment, however denominated, not to enforce a patent
 (such as an express permission to practice a patent or covenant not to
 sue for patent infringement).  To "grant" such a patent license to a
 party means to make such an agreement or commitment not to enforce a
 patent against the party.
  If you convey a covered work, knowingly relying on a patent license,
 and the Corresponding Source of the work is not available for anyone
 to copy, free of charge and under the terms of this License, through a
 publicly available network server or other readily accessible means,
 then you must either (1) cause the Corresponding Source to be so
 available, or (2) arrange to deprive yourself of the benefit of the
 patent license for this particular work, or (3) arrange, in a manner
 consistent with the requirements of this License, to extend the patent
 license to downstream recipients.  "Knowingly relying" means you have
 actual knowledge that, but for the patent license, your conveying the
 covered work in a country, or your recipient's use of the covered work
 in a country, would infringe one or more identifiable patents in that
 country that you have reason to believe are valid.
  If, pursuant to or in connection with a single transaction or
 arrangement, you convey, or propagate by procuring conveyance of, a
 covered work, and grant a patent license to some of the parties
 receiving the covered work authorizing them to use, propagate, modify
 or convey a specific copy of the covered work, then the patent license
 you grant is automatically extended to all recipients of the covered
 work and works based on it.
  A patent license is "discriminatory" if it does not include within
 the scope of its coverage, prohibits the exercise of, or is
 conditioned on the non-exercise of one or more of the rights that are
 specifically granted under this License.  You may not convey a covered
 work if you are a party to an arrangement with a third party that is
 in the business of distributing software, under which you make payment
 to the third party based on the extent of your activity of conveying
 the work, and under which the third party grants, to any of the
 parties who would receive the covered work from you, a discriminatory
 patent license (a) in connection with copies of the covered work
 conveyed by you (or copies made from those copies), or (b) primarily
 for and in connection with specific products or compilations that
 contain the covered work, unless you entered into that arrangement,
 or that patent license was granted, prior to 28 March 2007.
  Nothing in this License shall be construed as excluding or limiting
 any implied license or other defenses to infringement that may
 otherwise be available to you under applicable patent law.
  12. No Surrender of Others' Freedom.
  If conditions are imposed on you (whether by court order, agreement or
 otherwise) that contradict the conditions of this License, they do not
 excuse you from the conditions of this License.  If you cannot convey a
 covered work so as to satisfy simultaneously your obligations under this
 License and any other pertinent obligations, then as a consequence you may
 not convey it at all.  For example, if you agree to terms that obligate you
 to collect a royalty for further conveying from those to whom you convey
 the Program, the only way you could satisfy both those terms and this
 License would be to refrain entirely from conveying the Program.
  13. Use with the GNU Affero General Public License.
  Notwithstanding any other provision of this License, you have
 permission to link or combine any covered work with a work licensed
 under version 3 of the GNU Affero General Public License into a single
 combined work, and to convey the resulting work.  The terms of this
 License will continue to apply to the part which is the covered work,
 but the special requirements of the GNU Affero General Public License,
 section 13, concerning interaction through a network will apply to the
 combination as such.
  14. Revised Versions of this License.
  The Free Software Foundation may publish revised and/or new versions of
 the GNU General Public License from time to time.  Such new versions will
 be similar in spirit to the present version, but may differ in detail to
 address new problems or concerns.
  Each version is given a distinguishing version number.  If the
 Program specifies that a certain numbered version of the GNU General
 Public License "or any later version" applies to it, you have the
 option of following the terms and conditions either of that numbered
 version or of any later version published by the Free Software
 Foundation.  If the Program does not specify a version number of the
 GNU General Public License, you may choose any version ever published
 by the Free Software Foundation.
  If the Program specifies that a proxy can decide which future
 versions of the GNU General Public License can be used, that proxy's
 public statement of acceptance of a version permanently authorizes you
 to choose that version for the Program.
  Later license versions may give you additional or different
 permissions.  However, no additional obligations are imposed on any
 author or copyright holder as a result of your choosing to follow a
 later version.
  15. Disclaimer of Warranty.
  THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY
 APPLICABLE LAW.  EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT
 HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY
 OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO,
 THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
 PURPOSE.  THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM
 IS WITH YOU.  SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF
 ALL NECESSARY SERVICING, REPAIR OR CORRECTION.
  16. Limitation of Liability.
  IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING
 WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS
 THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY
 GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE
 USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF
 DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD
 PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS),
 EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF
 SUCH DAMAGES.
  17. Interpretation of Sections 15 and 16.
  If the disclaimer of warranty and limitation of liability provided
 above cannot be given local legal effect according to their terms,
 reviewing courts shall apply local law that most closely approximates
 an absolute waiver of all civil liability in connection with the
 Program, unless a warranty or assumption of liability accompanies a
 copy of the Program in return for a fee.
                     END OF TERMS AND CONDITIONS
            How to Apply These Terms to Your New Programs
  If you develop a new program, and you want it to be of the greatest
 possible use to the public, the best way to achieve this is to make it
 free software which everyone can redistribute and change under these terms.
  To do so, attach the following notices to the program.  It is safest
 to attach them to the start of each source file to most effectively
 state the exclusion of warranty; and each file should have at least
 the "copyright" line and a pointer to where the full notice is found.
    <one line to give the program's name and a brief idea of what it does.>
    Copyright (C) <year>  <name of author>
    This program is free software: you can redistribute it and/or modify
    it under the terms of the GNU General Public License as published by
    the Free Software Foundation, either version 3 of the License, or
    (at your option) any later version.
    This program is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
    GNU General Public License for more details.
    You should have received a copy of the GNU General Public License
    along with this program.  If not, see <https://www.gnu.org/licenses/>.
 Also add information on how to contact you by electronic and paper mail.
  If the program does terminal interaction, make it output a short
 notice like this when it starts in an interactive mode:
    <program>  Copyright (C) <year>  <name of author>
    This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
    This is free software, and you are welcome to redistribute it
    under certain conditions; type `show c' for details.
 The hypothetical commands `show w' and `show c' should show the appropriate
 parts of the General Public License.  Of course, your program's commands
 might be different; for a GUI interface, you would use an "about box".
  You should also get your employer (if you work as a programmer) or school,
 if any, to sign a "copyright disclaimer" for the program, if necessary.
 For more information on this, and how to apply and follow the GNU GPL, see
 <https://www.gnu.org/licenses/>.
  The GNU General Public License does not permit incorporating your program
 into proprietary programs.  If your program is a subroutine library, you
 may consider it more useful to permit linking proprietary applications with
 the library.  If this is what you want to do, use the GNU Lesser General
 Public License instead of this License.  But first, please read
 <https://www.gnu.org/licenses/why-not-lgpl.html>.
--- a/Documents/subject.pdf
+++ b/Documents/subject.pdf
--- a/Output/phase_separation_0.csv
+++ b/Output/phase_separation_0.csv
@@ -0,0 +1,200 @@
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--- a/Output/phase_separation_1.csv
+++ b/Output/phase_separation_1.csv
@@ -0,0 +1,200 @@
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--- a/Output/phase_separation_10.csv
+++ b/Output/phase_separation_10.csv
@@ -0,0 +1,200 @@
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--- a/Output/phase_separation_11.csv
+++ b/Output/phase_separation_11.csv
@@ -0,0 +1,200 @@
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--- a/Output/phase_separation_12.csv
+++ b/Output/phase_separation_12.csv
@@ -0,0 +1,200 @@
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--- a/Output/phase_separation_13.csv
+++ b/Output/phase_separation_13.csv
@@ -0,0 +1,200 @@
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--- a/Output/phase_separation_14.csv
+++ b/Output/phase_separation_14.csv
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--- a/Output/phase_separation_15.csv
+++ b/Output/phase_separation_15.csv
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--- a/Output/phase_separation_16.csv
+++ b/Output/phase_separation_16.csv
@@ -0,0 +1,200 @@
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--- a/Output/phase_separation_17.csv
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--- a/Output/phase_separation_18.csv
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--- a/Output/phase_separation_19.csv
+++ b/Output/phase_separation_19.csv
@@ -0,0 +1,200 @@
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--- a/Output/phase_separation_2.csv
+++ b/Output/phase_separation_2.csv
@@ -0,0 +1,200 @@
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--- a/Output/phase_separation_3.csv
+++ b/Output/phase_separation_3.csv
@@ -0,0 +1,200 @@
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--- a/Output/phase_separation_4.csv
+++ b/Output/phase_separation_4.csv
@@ -0,0 +1,200 @@
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--- a/Output/phase_separation_5.csv
+++ b/Output/phase_separation_5.csv
@@ -0,0 +1,200 @@
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--- a/Output/phase_separation_6.csv
+++ b/Output/phase_separation_6.csv
@@ -0,0 +1,200 @@
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--- a/Output/phase_separation_7.csv
+++ b/Output/phase_separation_7.csv
@@ -0,0 +1,200 @@
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--- a/Output/phase_separation_8.csv
+++ b/Output/phase_separation_8.csv
@@ -0,0 +1,200 @@
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--- a/Output/phase_separation_9.csv
+++ b/Output/phase_separation_9.csv
@@ -0,0 +1,200 @@
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 5.941879359715880419e-11
 4.420611657743980919e-07
 6.054815156827970523e-10
 5.213224593189643265e-04
 1.754444626263860444e-09
 2.621642289460840908e-10
 7.799735285117171560e-10
 1.676137224245580995e-09
 2.162297333704554187e-09
 1.205740114074660563e-09
 1.945308821496615703e-09
 4.821537542377593442e-09
 1.113034881908417194e-09
 2.915999295574257550e-10
 1.612403817110536310e-09
 3.228210201570732048e-09
 1.840162459044407539e-10
 6.152007022588388859e-10
 2.944306940803754709e-10
 6.981198010938709861e-10
 1.291700346454880760e-10
 2.179780083228090113e-10
 4.445753246151739048e-10
 6.197554472383654344e-10
 3.313834936554553491e-11
 1.547309746775029513e-10
 5.173059585822913636e-10
 8.547943068495513180e-11
 2.060815419793705799e-11
 3.893818853684848968e-10
 2.269366738165076359e-10
 1.095618135631012302e-09
 7.806271882711994635e-10
 4.471626142977932473e-10
 3.422638568336855482e-09
 7.836904116825822247e-10
 5.728203959291025968e-10
 1.355548322079322473e-09
 2.322199176558775284e-10
 3.844723091109726896e-11
 8.243898397321030772e-10
 1.614268163756810606e-10
 4.349657290970631387e-10
 5.348168277049025657e-10
 2.091138757534776922e-09
 2.832664658267289359e-10
 9.907273999467282747e-10
 1.806738524604745758e-10
 2.979350233086509111e-09
 1.600026002884663576e-10
 6.850488164895040481e-10
 1.828399020866276675e-09
 4.980629917892006775e-10
 1.241095053884285948e-09
 4.493955543109424494e-09
 2.331332210178815949e-10
 6.326632439126497285e-10
 8.888095049132318267e-11
 4.668533757067564912e-10
 6.268828525593555337e-11
 1.602243291448070461e-09
 8.709433287742407962e-10
 1.871862253866989882e-10
 2.133325359826320323e-09
 8.428743681543995734e-09
 1.928434028836529203e-10
 7.354440618037699865e-10
 1.479913533815996667e-09
 1.577130559087975963e-10
 2.709926202781955997e-11
 4.165241409835373021e-10
 6.612100737664327467e-10
 1.175697875617949132e-09
 2.515824408962527022e-09
 6.622154974216998755e-10
 1.382799117833331318e-10
 5.677854396470390236e-11
 5.401016829530785696e-09
 2.308812120469004268e-10
 8.389744898160989659e-09
 2.696895182623854330e-10
 6.154052021002337208e-11
 2.104062319905042009e-09
 2.522222422146752095e-09
 1.157159337433285153e-09
 4.105839232178712196e-10
 1.428368794815139825e-09
 3.843202549778969803e-10
 1.304570871111317279e-09
 3.164047014204028175e-10
 7.825194520566428692e-10
 2.996302724176090911e-10
 3.717552970036806275e-10
 1.643024872534784931e-09
 1.956030029970995579e-09
 1.473742258749153253e-10
 1.212904516931599877e-09
 6.845659008244848885e-11
 1.306867011438119436e-09
 3.671106768823839990e-10
 3.607645250358413205e-10
 1.271547304440260846e-09
 1.410476832145487633e-10
 8.777194348529978977e-08
 5.436595867603920012e-10
 5.392615887612488956e-11
 4.697923286840192622e-10
 1.395977983491569142e-09
 2.161970156880289840e-10
 3.202578844872048127e-10
 3.582482733634176229e-10
 1.032641571002874837e-10
 5.484900732732514326e-10
 2.042875950325382074e-09
 4.494310593158507387e-10
 6.009029069141305020e-11
 2.033340028314635281e-10
 1.059472633952315985e-08
 5.724119611359390478e-10
 4.262558919282174380e-10
 3.588028734573041541e-10
 1.034310158960732355e-09
 4.354956919192124779e-11
 5.667726317921485554e-10
 2.099662853222443629e-09
 1.751957409724426235e-10
 2.409911743727541883e-10
 1.926869601897814813e-10
 5.768792287312258741e-10
 8.134383783459529545e-10
 1.971793380911248667e-09
 6.395631224583827388e-09
 8.856713868640356596e-10
 2.938590882591995755e-10
 2.013951233437909510e-10
 3.516994383325869740e-10
 4.240894501293815024e-10
 1.078213993869413730e-09
 8.312236275374734306e-11
 2.538666646474372647e-09
 2.338729898287544851e-02
 5.300824096712303173e-12
 1.971658581841460717e-10
 3.658042167922816269e-11
 2.017693543911442578e-09
 3.972717942019663036e-09
 8.806378363003156635e-10
 8.457386574129837879e-11
 4.308278834371094448e-12
 3.178981892865428253e-10
 7.486326184905246461e-10
 2.055084022823571943e-09
 2.904875310377338078e-10
 1.543875668185210138e-10
 1.461186231324453012e-09
 4.353425989675314689e+00
 8.094978813709728829e-10
 2.337725281455106047e-10
 5.148059663550427355e-10
 9.847168843821252063e-10
 2.353595868336862623e-09
 6.923760288383933033e-10
 8.784160221347257952e-10
 8.819855307339105838e-11
 2.774590797407165433e-10
 1.694829529139946303e-10
 4.721913620929171185e-11
 1.071395823768589782e-08
 1.554956852076333563e-09
 3.114079287925047941e-10
 9.493009986532071995e-10
 3.262527292740506783e-10
 3.615515230268640232e-10
 2.128634709246496842e-10
 3.426583592024589521e-09
 1.627935988802954871e-09
 1.409925325731598413e-10
 1.344985369042397300e-11
 8.787480222312698231e-10
--- a/Output/poincare_sections_linear_10.csv
+++ b/Output/poincare_sections_linear_10.csv
--- a/Output/poincare_sections_linear_100.csv
+++ b/Output/poincare_sections_linear_100.csv
--- a/Output/poincare_sections_linear_12.csv
+++ b/Output/poincare_sections_linear_12.csv
--- a/Output/poincare_sections_linear_6.csv
+++ b/Output/poincare_sections_linear_6.csv
--- a/Output/poincare_sections_linear_8.csv
+++ b/Output/poincare_sections_linear_8.csv
--- a/Output/poincare_sections_parallel_10.csv
+++ b/Output/poincare_sections_parallel_10.csv
--- a/Output/poincare_sections_parallel_100.csv
+++ b/Output/poincare_sections_parallel_100.csv
--- a/Output/poincare_sections_parallel_12.csv
+++ b/Output/poincare_sections_parallel_12.csv
--- a/Output/poincare_sections_parallel_6.csv
+++ b/Output/poincare_sections_parallel_6.csv
--- a/Output/poincare_sections_parallel_8.csv
+++ b/Output/poincare_sections_parallel_8.csv
--- a/README.md
+++ b/README.md
@@ -0,0 +1,18 @@
 # Title
 Moussouni, Yaël (MSc student) & Bhat, Junaid Ramzan (MSc student)
 Université de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, 
 UMR 7550, F-67000, Strasbourg, France
 # Requirements
 # Installation
 # Usage
 ## Main Poincaré Sections (Linear and Parallel)
 ...
 The result of the simulation is saved in the Output directory, with the prefix `poincare_section_linear_` followed by 1/E (e.g. 12 for E = 1/12). The ASCII file contains all the Poincare section points (y on the first line, v on the second line) separated by blank spaces.
--- a/Source/TODEL_area.py
+++ b/Source/TODEL_area.py
@@ -0,0 +1,67 @@
 import numpy as np
 import matplotlib.pyplot as plt
 import potentials as pot
 import integrator as itg
 import initial_conditions as init
 # Parameters
 h = 0.01  # Step size for integrator
 N_inter = 500 # Number of iterations for distance calculation
 eps = 1e-3  # Perturbation distance
 mu_c = 0.6  # Threshold for classifying ergodic vs regular
 energy_values = np.array([0.01, 0.02, 0.04, 0.08, 0.10, 0.12, 0.14, 0.16, 0.18])
 N_PART = 100
 def classify_point(P1, P1_prime):
    pos_t1, positions1 = itg.rk4(0, P1, h, N_inter, pot.hh_evolution)
    pos_t2, positions2 = itg.rk4(0, P1_prime, h, N_inter, pot.hh_evolution)
    # Compute the sum of squared distances (μ)
    mu = 0
    for i in range(N_inter):
        y1, y_dot1 = positions1[i, 0, 1], positions1[i, 1, 1]
        y2, y_dot2 = positions2[i, 0, 1], positions2[i, 1, 1]
        x1, x_dot1 = positions1[i, 0, 0], positions1[i, 1, 0]
        x2, x_dot2 = positions2[i, 0, 0], positions2[i, 1, 0]
        squared_distance = (np.sqrt((x2 - x1)**2 + (y2 - y1) ** 2 + (x_dot2 - x_dot1) ** 2 +(y_dot2 - y_dot1) ** 2))**2
        mu += squared_distance
    return mu
 # Analyze for different energy values
 relative_areas = []
 for E in energy_values:
    # Generate initial conditions for the given energy level
    initial_conditions = init.n_energy_part(pot.hh_potential, N=N_PART, E=E)
    initial_conditions_eps = initial_conditions.copy()
    initial_conditions_eps[0,1] += eps
    n_regular = 0
    # Loop through each initial condition
    for i in range(N_PART):
        P1 = initial_conditions[:, :, i]
        # Perturb the initial point slightly
        P1_prime = initial_conditions_eps[:,:,i]
        # Classify the point based on the evolution
        mu = classify_point(P1, P1_prime)
        np.nan_to_num(mu, copy=False, nan=1.0)
        print(f"the mu value is {mu}")
        if mu < mu_c:
            n_regular += 1
    # Compute the relative area covered by regular points
    relative_area = n_regular / N_PART
    relative_areas.append(relative_area)
 # Plot the results
 plt.figure()
 plt.plot(energy_values, relative_areas, marker='o', color='C0')
 plt.xlabel("Energy (E)")
 plt.ylabel("Relative Area Covered by Regular Curves")
 plt.title("Transition from Ergodic to Stable Regime")
 plt.grid()
 plt.show()
--- a/Source/TODEL_equations.py
+++ b/Source/TODEL_equations.py
@@ -0,0 +1,17 @@
 from sympy import *
 # Forces
 X, Y, U, V = symbols("X, Y, U, V")
 POT = (X**2 + Y**2 + 2*X**2*Y - 2*Y**3/3)/2
 KIN = (U**2 + V**2)/2
 TOT = KIN + POT
 DX = diff(POT, X)
 DY = diff(POT, Y)
 FX = -DX
 FY = -DY
 print(FX)
 print(FY)
--- a/Source/TODO_chaos.py
+++ b/Source/TODO_chaos.py
@@ -0,0 +1,73 @@
 import numpy as np
 import matplotlib.pyplot as plt
 from scipy.integrate import quad
 import potentials as pot
 import integrator as itg
 import initial_conditions as init
 DEFAULT_N_iter = 30000
 DEFAULT_N_part = 100
 DEFAULT_h = 0.01
 DEFAULT_c = 1.7
 E_all = np.array([1/100, 1/12, 1/10, 1/8, 1/6])
 def compute_coordinates(E: float,
                        N_iter: int = DEFAULT_N_iter,
                        N_part: int = DEFAULT_N_part,
                        h: float = DEFAULT_h) -> tuple:
    W_part = init.n_energy_part(pot.hh_potential, N_part, E)
    # Perform integration
    t_part, coord_part = itg.rk4(0, W_part, h, N_iter, pot.hh_evolution)
    # Extract positions and velocities
    x_part = coord_part[:, 0, 0]
    y_part = coord_part[:, 0, 1]
    u_part = coord_part[:, 1, 0]
    v_part = coord_part[:, 1, 1]
    return t_part, x_part, y_part, u_part, v_part
 def x(t: float,
      x_part: np.ndarray,
      h: float = DEFAULT_h):
    return x_part[t//h]
 def theta(t: float, 
          x,
          x_part: np.ndarray,
          c: float = DEFAULT_c, 
          h: float = DEFAULT_h,
          ds: float = DEFAULT_h):
    """theta(t) = c*t + int_0^t phi(x(s)) ds"""
    I = quad(lambda s: x(s, x_part), 0, t)
    return c*t + I
 def p(t: float,
      x,
      theta,
      x_part: np.ndarray,
      h: float = DEFAULT_h,
      ds: float = DEFAULT_h):
    I = quad(lambda s: x(s, x_part)*cos(theta(s, x, x_part)), 0, t)
    return I
 def M(t,
      p,
      x_part: np.ndarray,
      h: float = DEFAULT_h,
      dtau: float = DEFAULT_h):
    I = quad(lambda tau: (p(t+tau, x, theta, x_part) 
                          - p(tau, x, theta, x_part)))**2, 0, np.max(t))
    return I
 if __name__ == "__main__":
    for i in range(len(E_all)):
        E = E_all[i]
        t_part, x_part, y_part, v_part, u_part = compute_coordinates(E)
        I = M(t_part, x_part)
--- a/Source/YII_1.mplstyle
+++ b/Source/YII_1.mplstyle
@@ -0,0 +1,78 @@
 ### Axes
 axes.titlesize : large
 axes.labelsize : large
 ### Lines
 lines.linewidth : 1.0
 hatch.linewidth: 0.25
 lines.markersize: 5
 patch.antialiased   : True
 ### Ticks
 xtick.top: True
 xtick.bottom: True
 xtick.major.size: 4.0
 xtick.minor.size: 2.0
 xtick.major.width: 0.5
 xtick.minor.width: 0.5
 xtick.direction: in
 xtick.minor.visible: True
 xtick.major.top: True
 xtick.major.bottom: True
 xtick.minor.top: True
 xtick.minor.bottom: True
 xtick.major.pad: 5.0
 xtick.minor.pad: 5.0
 ytick.labelsize: large
 ytick.left: True
 ytick.right: True
 ytick.major.size: 4.0
 ytick.minor.size: 2.0
 ytick.major.width: 0.5
 ytick.minor.width: 0.5
 ytick.direction: in
 ytick.minor.visible: True
 ytick.major.left: True
 ytick.major.right: True
 ytick.minor.left: True
 ytick.minor.right: True
 ytick.major.pad: 5.0
 ytick.minor.pad: 5.0
 xtick.labelsize: large
 ### Legend
 legend.frameon: True
 legend.fontsize: 9.0
 ### Figure size
 figure.figsize: 5.9, 3.9
 figure.subplot.left: 0.1
 figure.subplot.bottom: 0.175
 figure.subplot.top: 0.90
 figure.subplot.right: 0.95
 figure.autolayout: True
 figure.dpi: 150
 ### Fonts
 font.family: serif
 font.serif: Latin Modern Roman
 font.size: 9.0
 text.usetex: true
 ### Saving figures
 savefig.format: pdf
 savefig.dpi: 300
 savefig.pad_inches: 0
 path.simplify: True
 ### Cycler
 axes.prop_cycle: cycler(color=['06518E','E8BD0F','3B795E','ED1C24','C95BCF','009EDB'])
 ### Colormap
 image.cmap: cividis
 ### LaTeX 
 text.latex.preamble: \input{~/.config/matplotlib/stylelib/yii_plot.tex}
--- a/Source/YII_light_1.mplstyle
+++ b/Source/YII_light_1.mplstyle
@@ -0,0 +1,74 @@
 ### Axes
 axes.titlesize : large
 axes.labelsize : large
 ### Lines
 lines.linewidth : 1.0
 hatch.linewidth: 0.25
 lines.markersize: 5
 patch.antialiased   : True
 ### Ticks
 xtick.top: True
 xtick.bottom: True
 xtick.major.size: 4.0
 xtick.minor.size: 2.0
 xtick.major.width: 0.5
 xtick.minor.width: 0.5
 xtick.direction: in
 xtick.minor.visible: False
 xtick.major.top: True
 xtick.major.bottom: True
 xtick.minor.top: False
 xtick.minor.bottom: False
 xtick.major.pad: 5.0
 xtick.minor.pad: 5.0
 ytick.labelsize: large
 ytick.left: True
 ytick.right: True
 ytick.major.size: 4.0
 ytick.minor.size: 2.0
 ytick.major.width: 0.5
 ytick.minor.width: 0.5
 ytick.direction: in
 ytick.minor.visible: False
 ytick.major.left: True
 ytick.major.right: True
 ytick.minor.left: False
 ytick.minor.right: False
 ytick.major.pad: 5.0
 ytick.minor.pad: 5.0
 xtick.labelsize: large
 ### Legend
 legend.frameon: True
 legend.fontsize: 9.0
 ### Figure size
 figure.figsize: 5.9, 3.9
 figure.subplot.left: 0.1
 figure.subplot.bottom: 0.175
 figure.subplot.top: 0.90
 figure.subplot.right: 0.95
 figure.autolayout: True
 figure.dpi: 150
 ### Fonts
 font.family: monospace
 font.serif: Latin Modern Mono
 font.size: 9.0
 #text.usetex: true
 ### Saving figures
 savefig.format: pdf
 savefig.dpi: 300
 savefig.pad_inches: 0
 path.simplify: True
 ### Cycler
 axes.prop_cycle: cycler(color=['06518E','E8BD0F','3B795E','ED1C24','C95BCF','009EDB'])
 ### Colormap
 image.cmap: cividis
--- a/Source/energies.py
+++ b/Source/energies.py
@@ -0,0 +1,20 @@
 import numpy as np
 def kinetic(W: np.ndarray) -> np.ndarray:
    """Computes the kinetic energy.
    :param W: Phase-space vectors
    :returns T: Kinetic energy
    """
    U = W[1,0]
    V = W[1,1]
    # If U or V is not an array (or a list), but rather a scalar, then we
    # create a list of one element so that it can work either way
    if np.ndim(U) == 0: U = np.array([U])
    if np.ndim(V) == 0: V = np.array([V])
    return (U**2 + V**2)/2
 def total(W: np.ndarray,
          potential,
          kinetic = kinetic) -> np.ndarray:
    return potential(W) + kinetic(W)
--- a/Source/initial_conditions.py
+++ b/Source/initial_conditions.py
@@ -0,0 +1,96 @@
 import numpy as np
 import matplotlib.pyplot as plt
 POS_MIN = -1
 POS_MAX = +1
 VEL_MIN = -1
 VEL_MAX = +1
 N_PART = 1
 def mesh_grid(N: int = N_PART,
              xmin: float = POS_MIN, 
              xmax: float = POS_MAX, 
              ymin: float = POS_MIN, 
              ymax: float = POS_MAX) -> np.ndarray:
    """Generates a set of regularly sampled particles with no velocity
    :param N: number of particles
    :parma xmin: Minimum value for position x
    :param xmax: Maximum value for position x
    :param ymin: Minimum value for position y
    :param ymax: Maximum value for position y
    :returns W: Phase-space vector
    """
    X = np.linspace(xmin, xmax, N)
    Y = np.linspace(ymin, ymax, N)
    X,Y = np.meshgrid(X,Y, indexing="ij")
    return np.array([X,Y])
 def one_part(x0: float = 0, 
             y0: float = 0, 
             u0: float= 0, 
             v0: float = 0) -> np.ndarray:
    """Generates a particle at position (x0, y0) 
    with velocities (u0,v0)
    :param x0: Initial position x
    :param y0: Initial position y
    :param u0: Initial velocity u
    :param v0: Initial velocity v
    """
    X = x0
    Y = y0
    U = u0
    V = v0
    return np.array([[X,Y], [U,V]])
 def n_energy_part(potential,
                  N: int = N_PART,
                  E: float = 0,
                  xmin: float = -1,
                  xmax: float = +1,
                  ymin: float = -0.5,
                  ymax: float = +1):
    X = []
    Y = []
    POT = []
    U = []
    V = []
    while len(X) < N:
        x = np.random.random()*(xmax-xmin)+xmin
        y = np.random.random()*(ymax-ymin)+ymin
        w = np.array([x, y])
        pot = potential(w, position_only=True)[0]
        if pot <= E:
            X.append(x)
            Y.append(y)
            POT.append(pot)
    X = np.array(X)
    Y = np.array(Y)
    POT = np.array(POT)
    U = np.zeros_like(X)
    V = np.zeros_like(Y)
    C = np.sqrt(2 * (E - POT))
    THETA = np.random.random(N)*2*np.pi
    U = C*np.cos(THETA)
    V = C*np.sin(THETA)
    return np.array([[X, Y], [U, V]])
 def n_energy_2part(potential,
                   N: int = N_PART,
                   E: float = 0,
                   sep: float = 1e-7,
                   xmin: float = -1,
                   xmax: float = +1,
                   ymin: float = -0.5,
                   ymax: float = +1):
    W_1 = n_energy_part(potential, N, E)
    W_2 = np.zeros_like(W_1)
    alpha = np.random.uniform(0, 2*np.pi, N)
    W_2[0, 0] = W_1[0, 0] + sep*np.cos(alpha)
    W_2[0, 1] = W_1[0, 1] + sep*np.sin(alpha)
    W_2[1, 0] = W_1[1, 0]
    W_2[1, 1] = W_1[1, 1]
    return (W_1, W_2)
--- a/Source/integrator.py
+++ b/Source/integrator.py
@@ -0,0 +1,87 @@
 import numpy as np
 def euler(x0, y0, h, n, func): # FIXME cannot be used with vectors
    """
    Euler method adapted for state vector [[x, y], [u, v]]
    :param x0: initial time value
    :param y0: initial state vector [[x, y], [u, v]]
    :param h: step size (time step)
    :param n: number of steps
    :param func: RHS of differential equation
    :returns: x array, solution array
    """
    x_values = np.zeros(n)
    y_values = np.zeros((n, 2, 2))  #  to accommodate the state vector
    for i in range(n):
        dydt = func(x0, y0)
        y0 = y0 + h * dydt
        x0 = x0 + h
        x_values[i] = x0
        y_values[i, :, :] = y0
    return x_values, y_values
 # Updated RK2 integrator
 def rk2(x0, y0, h, n, func): # FIXME cannot be used with vectors
    """
    RK2 method adapted for state vector [[x, y], [u, v]]
    :param x0: initial time value
    :param y0: initial state vector [[x, y], [u, v]]
    :param h: step size (time step)
    :param n: number of steps
    :param func: RHS of differential equation
    :returns: x array, solution array
    """
    x_values = np.zeros(n)
    y_values = np.zeros((n, 2, 2))  #  to accommodate the state vector
    for i in range(n):
        k1 = func(x0, y0)
        k2 = func(x0 + h / 2., y0 + h / 2. * k1)
        y0 = y0 + h * (k1 / 2. + k2 / 2.)
        x0 = x0 + h
        x_values[i] = x0
        y_values[i, :, :] = y0
    return x_values, y_values
 def rk4(t0: float, 
        W0: np.ndarray, 
        h: float, 
        n: int, 
        func):
    """
    RK4 method adapted for state vector [[x, y], [u, v]]
    :param x0: initial time value
    :param y0: initial state vector [[x, y], [u, v]]
    :param h: step size (time step)
    :param n: number of steps
    :param func: RHS of differential equation
    :returns: x array, solution array
    """
    time = np.zeros(n)
    W = np.zeros((n,) + np.shape(W0))
    # to accommodate the state vector
    t = t0
    w = W0
    for i in range(n):
        k1 = func(t, w)
        k2 = func(t + h / 2., w + h / 2. * k1)
        k3 = func(t + h / 2., w + h / 2. * k2)
        k4 = func(t + h, w + h * k3)
        w = w + h * (k1 / 6. + k2 / 3. + k3 / 3. + k4 / 6.)
        t = t + h
        time[i] = t
        W[i] = w
    return t, W
 def integrator_type(x0, y0, h, n, func, int_type):
    return int_type(x0, y0, h, n, func)
--- a/Source/main_area.py
+++ b/Source/main_area.py
@@ -0,0 +1,70 @@
 #!/usr/bin/env python
 """
 Main: Compute Relative Area
 Computes the relative area covered bu the curves for different energies, to 
 study ordered and chaotic regimes.
@ Author: Moussouni, Yaël (MSc student) & Bhat, Junaid Ramzan (MSc student)
@ Institution:  Université de Strasbourg, CNRS, Observatoire astronomique
                de Strasbourg, UMR 7550, F-67000 Strasbourg, France
@ Date: 2024-12-13
 """
 import numpy as np
 from scipy.optimize import curve_fit
 import potentials as pot
 import energies as ene
 import integrator as itg
 import initial_conditions as init
 import poincare_sections as pcs
 OUT_DIR = "./Output/"
 FILENAME_PREFIX = "phase_separation_"
 EXTENSION = ".csv"
 DEFAULT_N_iter = int(1e5)
 DEFAULT_N_part = 200
 DEFAULT_h = 0.005
 E_all = np.linspace(1/100, 1/6, 20)
 def compute_mu(E: float,
               N_iter: int = DEFAULT_N_iter,
               N_part: int = DEFAULT_N_part,
               h: float = DEFAULT_h) -> tuple:
    """
    Computes the phase-space squared distances for particles of given energy E.
    @params:
        - E: the total energy of each particles
        - N_iter: the number of iteration
        - N_part: the number of particles
        - h: integration steps
    @returns:
        - mu: phase-space squared distance
    """
    W_1, W_2 = init.n_energy_2part(pot.hh_potential, N_part, E)
    t_1, positions_1 = itg.rk4(0, W_1, h, N_iter, pot.hh_evolution)
    x_1 = positions_1[:, 0, 0]
    y_1 = positions_1[:, 0, 1]
    u_1 = positions_1[:, 1, 0]
    v_1 = positions_1[:, 1, 1]
    t_2, positions_2 = itg.rk4(0, W_2, h, N_iter, pot.hh_evolution)
    x_2 = positions_2[:, 0, 0]
    y_2 = positions_2[:, 0, 1]
    u_2 = positions_2[:, 1, 0]
    v_2 = positions_2[:, 1, 1]
    dist_sq = (x_2[-25:] - x_1[-25:])**2 \
            + (y_2[-25:] - y_1[-25:])**2 \
            + (u_2[-25:] - u_1[-25:])**2 \
            + (v_2[-25:] - v_1[-25:])**2
    mu = np.sum(dist_sq, axis=0)
    return mu
 if __name__ == "__main__":
    mu_all = []
    for i in range(len(E_all)):
        mu = compute_mu(E_all[i])
        filename = OUT_DIR + FILENAME_PREFIX\
                 + str(i) + EXTENSION
        np.savetxt(filename, mu)
--- a/Source/main_poincare_sections_linear.py
+++ b/Source/main_poincare_sections_linear.py
@@ -0,0 +1,86 @@
 #!/usr/bin/env python
 """
 Main: Computes Poincaré Sections (Linear Algorithm)
 Computes the Poincaré Sections with a linear algorithm 
 (i.e. no parallel computing).
@ Author: Moussouni, Yaël (MSc student) & Bhat, Junaid Ramzan (MSc student)
@ Institution:  Université de Strasbourg, CNRS, Observatoire astronomique
                de Strasbourg, UMR 7550, F-67000 Strasbourg, France
@ Date: 2024-11-29
 """
 import numpy as np
 import potentials as pot
 import integrator as itg
 import initial_conditions as init
 import poincare_sections as pcs
 # Parameters
 OUT_DIR = "./Output/"
 FILENAME_PREFIX = "poincare_sections_linear_"
 EXTENSION = ".csv"
 DEFAULT_N_iter = 30000
 DEFAULT_N_part = 100
 DEFAULT_h = 0.01
 E_all = np.array([1/100, 1/12, 1/10, 1/8, 1/6])
 text_E = ["1/100", "1/12", "1/10", "1/8", "1/6"]
 def compute_poincare_sections_linear(E: float,
                                    N_iter: int = DEFAULT_N_iter,
                                    N_part: int = DEFAULT_N_part,
                                    h: float = DEFAULT_h) -> tuple:
    """
    Computes the Poincaré sections for a given energy E.
    @params:
        - E: the total energy of each particles
        - N_iter: the number of iteration
        - N_part: the number of particles
        - h: integration steps
    @returns:
        - y_section, v_section: arrays containing the y and v coordinates of 
          the Poincaré sections
    """
    W_all_part = init.n_energy_part(pot.hh_potential, N_part, E)
    y_section = []
    v_section = []
    for i in range(N_part):
        W_part = W_all_part[:,:,i]
        # Perform integration
        t_part, coord_part = itg.rk4(0, W_part, h, N_inter, pot.hh_evolution)
        # Extract positions and velocities
        x_part = coord_part[:, 0, 0]
        y_part = coord_part[:, 0, 1]
        u_part = coord_part[:, 1, 0]
        v_part = coord_part[:, 1, 1]
        # Find Poincaré section points for the current initial condition
        y_pcs, v_pcs = pcs.pcs_find_legacy(x_part, y_part, u_part, v_part)
        # The legacy is important here, the algorithm is the same but the 
        # data format is different...
        # Append the current Poincaré section points to the overall lists
        y_section += y_pcs
        v_section += v_pcs
    return y_section, v_section
 if __name__ == "__main__":
    y_section_all = []
    v_section_all = []
    for i in range(len(E_all)):
        y_section, v_section = compute_poincare_sections_linear(E_all[i])
        section = np.array([y_section, v_section])
        filename = OUT_DIR + FILENAME_PREFIX\
                + str(text_E[i][2:]) + EXTENSION
        np.savetxt(filename, section)
--- a/Source/main_poincare_sections_parallel.py
+++ b/Source/main_poincare_sections_parallel.py
@@ -0,0 +1,71 @@
 #!/usr/bin/env python
 """
 Main: Computes Poincaré Sections (Parallel Algorithm)
 Computes the Poincaré Sections with a parallel algorithm (with Numpy)
@ Author: Moussouni, Yaël (MSc student) & Bhat, Junaid Ramzan (MSc student)
@ Institution:  Université de Strasbourg, CNRS, Observatoire astronomique
                de Strasbourg, UMR 7550, F-67000 Strasbourg, France
@ Date: 2024-11-29
 """
 import numpy as np
 import potentials as pot
 import integrator as itg
 import initial_conditions as init
 import poincare_sections as pcs
 # Parameters
 OUT_DIR = "./Output/"
 FILENAME_PREFIX = "poincare_sections_parallel_"
 EXTENSION = ".csv"
 DEFAULT_N_iter = 30000
 DEFAULT_N_part = 100
 DEFAULT_h = 0.01
 E_all = np.array([1/100, 1/12, 1/10, 1/8, 1/6])
 text_E = ["1/100", "1/12", "1/10", "1/8", "1/6"]
 def compute_poincare_sections_numpy(E: float,
                                    N_iter: int = DEFAULT_N_iter,
                                    N_part: int = DEFAULT_N_part,
                                    h: float = DEFAULT_h) -> tuple:
    """
    Computes the Poincaré sections for a given energy E.
    @params:
        - E: the total energy of each particles
        - N_iter: the number of iteration
        - N_part: the number of particles
        - h: integration steps
    @returns:
        - y_section, v_section: arrays containing the y and v coordinates of 
          the Poincaré sections
    """
    W_part = init.n_energy_part(pot.hh_potential, N_part, E)
    y_section = []
    v_section = []
    # Perform integration
    t_part, coord_part = itg.rk4(0, W_part, h, N_iter, pot.hh_evolution)
    # Extract positions and velocities
    x_part = coord_part[:, 0, 0]
    y_part = coord_part[:, 0, 1]
    u_part = coord_part[:, 1, 0]
    v_part = coord_part[:, 1, 1]
    # Find Poincaré section points for the current initial condition
    y_section, v_section = pcs.pcs_find(x_part, y_part, u_part, v_part)
    return y_section, v_section
 if __name__ == "__main__":
    y_section_all = []
    v_section_all = []
    for i in range(len(E_all)):
        y_section, v_section = compute_poincare_sections_numpy(E_all[i])
        section = np.array([y_section, v_section])
        filename = OUT_DIR + FILENAME_PREFIX\
                 + str(text_E[i][2:]) + EXTENSION
        np.savetxt(filename, section)
--- a/Source/main_yael.py
+++ b/Source/main_yael.py
@@ -0,0 +1,114 @@
 import time
 import numpy as np
 import matplotlib.pyplot as plt
 from scipy.optimize import curve_fit
 import potentials as pot
 import energies as ene
 import integrator as itg
 import initial_conditions as init
 import poincare_sections as pcs
 if "YII_light_1" in plt.style.available: 
    plt.style.use("YII_light_1")
 # Loading matplotlib style...
 # PARAM
 N_grid = 1000
 N_inter = int(1e5)
 N_part = 200
 #all_E = np.array([1/100, 1/12, 1/10, 1/8, 1/6])
 all_E = np.linspace(1/100, 1/6, 20)
 h = 0.005
 mu_c = 1e-4
 d_12 = 1e-7
 # INIT
 W_grid = init.mesh_grid(N_grid, xmin=-1, xmax=1, ymin=-1, ymax=1)
 X_grid = W_grid[0]
 Y_grid = W_grid[1]
 potential = pot.hh_potential(W_grid, position_only=True)
 # MAIN
 def gen_mu(E):
    pot_valid = np.ma.masked_where(potential > E, potential)
    W_1 = init.n_energy_part(pot.hh_potential, N_part, E)
    W_2 = np.zeros_like(W_1)
    alpha = np.random.uniform(0, 2*np.pi, N_part)
    W_2[0, 0] = W_1[0, 0] + d_12*np.cos(alpha)
    W_2[0, 1] = W_1[0, 1] + d_12*np.sin(alpha)
    W_2[1, 0] = W_1[1, 0]
    W_2[1, 1] = W_1[1, 1]
    t_1, positions_1 = itg.rk4(0, W_1, h, N_inter, pot.hh_evolution)
    x_1 = positions_1[:, 0, 0]
    y_1 = positions_1[:, 0, 1]
    u_1 = positions_1[:, 1, 0]
    v_1 = positions_1[:, 1, 1]
    t_2, positions_2 = itg.rk4(0, W_2, h, N_inter, pot.hh_evolution)
    x_2 = positions_2[:, 0, 0]
    y_2 = positions_2[:, 0, 1]
    u_2 = positions_2[:, 1, 0]
    v_2 = positions_2[:, 1, 1]
    dist_sq = (x_2[-25:] - x_1[-25:])**2 \
            + (y_2[-25:] - y_1[-25:])**2 \
            + (u_2[-25:] - u_1[-25:])**2 \
            + (v_2[-25:] - v_1[-25:])**2
    mu = np.sum(dist_sq, axis=0)
    return mu
 A = np.zeros_like(all_E)
 MU = []
 ALL_E = []
 for i in range(len(all_E)):
    mu = gen_mu(all_E[i])
    n_total = N_part
    n_ergo = np.sum(mu < mu_c) # count how many times the condition is verified
    A[i] = n_ergo/n_total
    MU.append(mu)
    ALL_E.append([all_E[i]]*len(mu))
 MU = np.array(MU)
 ALL_E = np.array(ALL_E)
 # def lin(x, a, b):
 #     return a*x + b
 # w_chaos = np.argwhere(A < 1).flatten()
 # X = all_E[w_chaos]
 # Y = A[w_chaos]
 # popt, pcov = curve_fit(lin, X, Y)
 # a, b = popt
 # da, db = np.sqrt(np.diag(pcov))
 fig, ax = plt.subplots(1)
 ax.scatter(ALL_E, MU, s=1, 
           color="k", alpha=0.1, label="Data")
 ax.scatter(all_E, np.mean(MU, axis=1), s=10, 
           color="C3", marker="*", label="Mean")
 ax.scatter(all_E, np.median(MU, axis=1), s=10, 
           color="C2", marker="s", label="Median")
 ax.hlines(mu_c, np.min(all_E), np.max(all_E), color="C4")
 ax.legend()
 ax.set_yscale("log")
 ax.set_xlabel("energy E")
 ax.set_ylabel("quantity mu")
 fig.tight_layout()
 fig.savefig("mu")
 fig, ax = plt.subplots(1)
 ax.scatter(all_E, A, color="C0", s=5, marker="o", label="Data")
 ax.set_xlabel("energy E")
 ax.set_ylabel("area A")
 ax.legend()
 fig.tight_layout()
 fig.savefig("area")
 plt.show()
--- a/Source/plot_poincare_sections.py
+++ b/Source/plot_poincare_sections.py
@@ -0,0 +1,88 @@
 #!/usr/bin/env python
 """
 Plot: Poincaré Sections (Linear and Parallel)
 Plots the Poincaré sections for different energies, computed either with linear
 or parallel algorithms.
@ Author: Moussouni, Yaël (MSc student) & Bhat, Junaid Ramzan (MSc student)
@ Institution:  Université de Strasbourg, CNRS, Observatoire astronomique
                de Strasbourg, UMR 7550, F-67000 Strasbourg, France
@ Date: 2024-11-29
 """
 import os
 import numpy as np
 import matplotlib.pyplot as plt
 if "YII_1" in plt.style.available: plt.style.use("YII_1")
 OUT_DIR = "./Output/"
 FILENAME_PREFIX = "poincare_sections_"
 EXTENSION = ".csv"
 def plot_poincare_sections(filelist: list, title:str = "") -> int:
    """
    Plot all the Poincaré sections in the file list.
    @params:
        - filelist: the list of files in the output directory, with the format 
        "poincare_sections_{linear, parallel}_[1/E].csv"
        - title: title of the figure
    @returns: 
        - 0.
    """
    orderlist = np.argsort([(int(file
                                 .replace(FILENAME_PREFIX, "")
                                 .replace(EXTENSION, "")
                                 .replace("linear_", "")
                                 .replace("parallel_", ""))) 
                            for file in filelist])
    filelist = np.array(filelist)[orderlist]
    N = len(filelist)
    fig, axs = plt.subplot_mosaic("ABC\nDEF")
    axs = list(axs.values())
    fig.suptitle(title)
    for i in range(N):
        ax = axs[i]
        filename = filelist[i]
        inv_E = (filename
                 .replace(FILENAME_PREFIX, "")
                 .replace(EXTENSION, "")
                 .replace("linear_", "")
                 .replace("parallel_", ""))
        data = np.loadtxt(OUT_DIR + filename)
        y_section = data[0]
        v_section = data[1]
        ax.scatter(y_section, v_section, 
                   s=.1, color="C3", marker=",", alpha=0.5,
                   label="$E = 1/{}$".format(inv_E))
        ax.set_xlabel("$y$")
        ax.set_ylabel("$v$")
        ax.legend(loc="upper right")
    while i < N:
        i += 1
        axs[i].axis('off')
    return 0
 print("\033[32m" 
      + "[P]arallel or [L]inear algorithm result, or [B]oth?" 
      + "\033[0m")
 answer = input("\033[32m" + "> " + "\033[0m").upper()
 if answer == "P":
    FILENAME_PREFIX += "parallel_"
 elif answer == "L":
    FILENAME_PREFIX += "linear_"
 filelist = [fname for fname in os.listdir(OUT_DIR) if FILENAME_PREFIX in fname]
 if answer in ["L", "B"]:
    filelist_linear = [fname for fname in filelist if "linear_" in fname]
    plot_poincare_sections(filelist_linear, 
                           title=("Poincaré Sections "
                                  "(results from the linear algorithm)"))
 if answer in ["P", "B"]:
    filelist_parallel = [fname for fname in filelist if "parallel_" in fname]
    plot_poincare_sections(filelist_parallel, 
                           title=("Poincaré Sections "
                                  "(results from the parallel algorithm)"))
 plt.show()
--- a/Source/poincare_sections.py
+++ b/Source/poincare_sections.py
@@ -0,0 +1,55 @@
 import numpy as np
 def pcs_find(pos_x, pos_y, vel_x, vel_y):
    if np.ndim(pos_x) == 1: 
        pos_x = np.array([pos_x])
        pos_y = np.array([pos_y])
        vel_x = np.array([vel_x])
        vel_y = np.array([vel_y])
    pcs_pos_y = []
    pcs_vel_y = []
    for j in range(len(pos_x[0])): # for each particle
        i = 0
        x = pos_x[:,j]
        y = pos_y[:,j]
        u = vel_x[:,j]
        v = vel_y[:,j]
        while i < len(x) - 1:
            if x[i] * x[i+1] < 0:
                y0 = y[i] \
                        + (y[i+1] - y[i])/(x[i+1] - x[i]) \
                        * (0 - x[i])
                v0 = v[i] \
                        + (v[i+1] - v[i])/(x[i+1] - x[i])\
                        * (0 - x[i])
                pcs_pos_y.append(y0)
                pcs_vel_y.append(v0)
            i += 1
    return pcs_pos_y, pcs_vel_y
 def pcs_find_legacy(pos_x, pos_y, vel_x, vel_y):
    """Depreciated legacy function that should not be used
    """
    if np.ndim(pos_x) == 1: 
        pos_x = np.array([pos_x])
        pos_y = np.array([pos_y])
        vel_x = np.array([vel_x])
        vel_y = np.array([vel_y])
    pcs_pos_y = []
    pcs_vel_y = []
    for j in range(len(pos_x)): # for each particle
        i = 0
        x = pos_x[j]
        y = pos_y[j]
        u = vel_x[j]
        v = vel_y[j]
        while i < len(x) - 1:
            if x[i] * x[i+1] < 0:
                y0 = y[i] \
                        + (y[i+1] - y[i])/(x[i+1] - x[i]) \
                        * (0 - x[i])
                v0 = v[i] \
                        + (v[i+1] - v[i])/(x[i+1] - x[i])\
                        * (0 - x[i])
                pcs_pos_y.append(y0)
                pcs_vel_y.append(v0)
            i += 1
    return pcs_pos_y, pcs_vel_y
--- a/Source/poincare_test.py
+++ b/Source/poincare_test.py
@@ -0,0 +1,58 @@
 import numpy as np
 import matplotlib.pyplot as plt
 import potentials as pot
 import energies as ene
 import integrator as itg
 import initial_conditions as init
 import poincare_sections as pcs
 h = 0.01
 N_inter = 30000
 eps = 1e-2
 x_0 = 0.0
 y_vals = np.linspace(0, 0.6, 50)
 E = 0.125  # 1/12
 # Initialize lists to store all Poincaré points across different initial conditions
 all_pcs_pos_y = []
 all_pcs_vel_y = []
 for i in range(len(y_vals)):
    y0 = y_vals[i]
    W_part = init.one_part(x_0, y0, 0, 0)
    POT = pot.hh_potential(W_part)[0]
    u_0 = np.sqrt(2 * (E - POT))
    v_0 = 0.0
    W_part[1, 0] = u_0
    W_part[1, 1] = v_0
    # Perform integration
    pos_t, positions = itg.rk4(0, W_part, h, N_inter, pot.hh_evolution)
    # Extract positions and velocities
    pos_x = positions[:, 0, 0]
    pos_y = positions[:, 0, 1]
    vel_x = positions[:, 1, 0]
    vel_y = positions[:, 1, 1]
    # Find Poincaré section points for the current initial condition
    pcs_pos_y, pcs_vel_y = pcs.pcs_find(pos_x, pos_y, vel_x, vel_y)
    # Append the current Poincaré section points to the overall lists
    all_pcs_pos_y.extend(pcs_pos_y)
    all_pcs_vel_y.extend(pcs_vel_y)
 # Plotting all Poincaré section points on the same graph
 fig, ax = plt.subplots()
 ax.scatter(all_pcs_pos_y, all_pcs_vel_y, s=2, color="C4")
 # Set labels and legend
 ax.set_xlabel("coordinate y")
 ax.set_ylabel("velocity v")
 ax.set_title("Combined Poincaré Section for Multiple Initial y0 Values")
 plt.show()
--- a/Source/potentials.py
+++ b/Source/potentials.py
@@ -0,0 +1,65 @@
 import numpy as np
 MAX_VAL = 1e3
 def kepler_potential(W_grid: np.ndarray, 
                     position_only: bool = False) -> np.ndarray:
    """Computes the Kepler potential: V(R) = -G*m1*m2/R 
    (assuming G = 1, m1 = 1, m2 = 1)
    assuming the point mass at (x = 0, y = 0).
    @params:
        - W: Phase-space vector
        - position_only: True if W is np.array([X, Y])
    @returns:
        - V: computed potential
    """
    if position_only: 
        X = W_grid[0]
        Y = W_grid[1]
    else:
        X = W_grid[0,0]
        Y = W_grid[0,1]
    # If X or Y is not an array (or a list), but rather a scalar, then we
    # create a list of one element so that it can work either way
    if np.ndim(X) == 0: X = np.array([X])
    if np.ndim(Y) == 0: Y = np.array([Y])
    R = np.sqrt(X**2 + Y**2)
    return -1/R
 def hh_potential(W_grid: np.ndarray, 
                 position_only=False) -> np.ndarray:
    """Computes the Hénon-Heiles potential.
    :param W: Phase-space vector
    :output V: Potential
    """
    if position_only:
        X = W_grid[0]
        Y = W_grid[1]
    else:
        X = W_grid[0, 0]
        Y = W_grid[0, 1]
    # If X or Y is not an array (or a list), but rather a scalar, then we
    # create a list of one element so that it can work either way
    if np.ndim(X) == 0: X = np.array([X])
    if np.ndim(Y) == 0: Y = np.array([Y])
    POT = (X**2 + Y**2 + 2*X**2*Y - 2*Y**3/3)/2
    return POT        
 def hh_evolution(t: np.ndarray, W: np.ndarray):
    """Computes the evolution from the HH potential
    :param t: Time (not used)
    :param W: Phase space vector
    :returns dot W: Time derivative of the phase space vector
    """
    X = W[0 ,0]
    Y = W[0, 1]
    U = W[1, 0]
    V = W[1, 1]
    DX = U 
    DY = V
    DU = -(2*X*Y + X)
    DV = -(X**2 - Y**2 + Y)
    return np.array([[DX, DY], [DU, DV]])
--- a/Source/test_evolution.py
+++ b/Source/test_evolution.py
@@ -0,0 +1,69 @@
 #!/usr/bin/env python
 """
 Test: Evolution
 Evolve a particle with a given energy in a potential and show the result path
 followed by the particle in a given time.
@ Author: Moussouni, Yaël (MSc student) & Bhat, Junaid Ramzan (MSc student)
@ Institution:  Université de Strasbourg, CNRS, Observatoire astronomique
                de Strasbourg, UMR 7550, F-67000 Strasbourg, France
@ Date: 2024-11-29
 """
 import numpy as np
 import matplotlib.pyplot as plt
 import potentials as pot
 import energies as ene
 import integrator as itg
 import initial_conditions as init
 import poincare_sections as pcs
 if "YII_1" in plt.style.available: plt.style.use("YII_1")
 # Loading matplotlib style...
 # Constants
 N_grid = 5000
 h = 0.01
 N_inter = 5000
 eps = 1e-2
 x_0 = 0.0
 y_0 = 0.5
 E = 0.125
 # Main
 if __name__ == "__main__":
    W_part = init.one_part(x_0, y_0,0,0)
    POT = pot.hh_potential(W_part)[0]
    u_0 = np.sqrt(2 * (E - POT))
    v_0 = 0.0
    W_part[1,0] = u_0
    W_part[1,1] = v_0
    pos_t, positions = itg.rk4(0, W_part, h, N_inter, pot.hh_evolution)
    pos_x = positions[:,0,0]
    pos_y = positions[:,0,1]
    vel_x = positions[:,1,0]
    vel_y = positions[:,1,1]
    W_grid = init.mesh_grid(N_grid)
    X_grid = W_grid[0]
    Y_grid = W_grid[1]
    potential = pot.hh_potential(W_grid, position_only=True)
    fig, ax = plt.subplots(1)
    pcm = ax.pcolormesh(X_grid, Y_grid, potential)
    line = ax.plot(pos_x, pos_y, color="C3")
    fig.colorbar(pcm, label="potential")
    ax.set_xlabel("$x$")
    ax.set_ylabel("$y$")
    ax.set_aspect("equal")
    plt.show(block=True)
--- a/Source/test_initial_E.py
+++ b/Source/test_initial_E.py
@@ -0,0 +1,53 @@
 #!/usr/bin/env python
 """
 Test: Initial Conditions With a Given Energy
 Sample random particles with the same given energy in a valid coordinates range.
@ Author: Moussouni, Yaël (MSc student) & Bhat, Junaid Ramzan (MSc student)
@ Institution:  Université de Strasbourg, CNRS, Observatoire astronomique
                de Strasbourg, UMR 7550, F-67000 Strasbourg, France
@ Date: 2024-11-29
 """
 import numpy as np
 import matplotlib.pyplot as plt
 import potentials as pot
 import energies as ene
 import integrator as itg
 import initial_conditions as init
 import poincare_sections as pcs
 if "YII_1" in plt.style.available: plt.style.use("YII_1")
 # Loading matplotlib style...
 # parameters
 N_grid = 1000
 N_inter = 30000
 N_part = 100
 E = 1/12
 h = 0.01
 if __name__ == "__main__":
    # Initial conditions
    W_grid = init.mesh_grid(N_grid, xmin=-1, xmax=1, ymin=-1, ymax=1)
    X_grid = W_grid[0]
    Y_grid = W_grid[1]
    potential = pot.hh_potential(W_grid, position_only=True)
    pot_valid = np.ma.masked_where(potential > E, potential)
    W_all_part = init.n_energy_part(pot.hh_potential, N_part, E)
    # Plot
    fig, ax = plt.subplots(1)
    pcm = ax.pcolormesh(X_grid, Y_grid, pot_valid, vmin = 0)
    sct = ax.scatter(W_all_part[0, 0], W_all_part[0, 1], s=1, color="C3")
    fig.colorbar(pcm, label="potential")
    ax.set_title("$E = {:.2f}$".format(E))
    ax.set_xlabel("$x$")
    ax.set_ylabel("$y$")
    ax.set_aspect("equal")
    plt.show(block=True)
--- a/Source/test_potentials.py
+++ b/Source/test_potentials.py
@@ -0,0 +1,56 @@
 #!/usr/bin/env python
 """
 Test: Potential
 Draw the Kepler potential and the Hénon--Heils potential
@ Author: Moussouni, Yaël (MSc student) & Bhat, Junaid Ramzan (MSc student)
@ Institution:  Université de Strasbourg, CNRS, Observatoire astronomique
                de Strasbourg, UMR 7550, F-67000 Strasbourg, France
@ Date: 2024-11-29
 """
 import numpy as np
 import matplotlib.pyplot as plt
 import potentials as pot
 import initial_conditions as init
 if "YII_1" in plt.style.available: plt.style.use("YII_1")
 # Loading matplotlib style...
 W = init.mesh_grid(300)
 def kepler(W):
    """Plots the Kepler potential"""
    X = W[0]
    Y = W[1]
    POT = pot.kepler_potential(W, position_only=True)
    fig, ax = plt.subplots(1)
    ax.set_title("Kepler potential")
    pcm = ax.pcolormesh(X, Y, POT)
    fig.colorbar(pcm, label="potential")
    ax.set_aspect("equal")
    ax.set_xlabel("$x$")
    ax.set_ylabel("$y$")
    return 0
 def hh(W):
    """Plots the Hénon--Heils potential"""
    X = W[0]
    Y = W[1]
    POT = pot.hh_potential(W, position_only=True)
    fig, ax = plt.subplots(1)
    ax.set_title("Hénon–Heils potential")
    pcm = ax.pcolormesh(X, Y, POT)
    fig.colorbar(pcm, label="potential")
    ax.set_aspect("equal")
    ax.set_xlabel("$x$")
    ax.set_ylabel("$y$")
    return 0
 if __name__ == "__main__":
    kepler(W)
    hh(W)
    plt.show(block=True)
--- a/Source/time_poincare_sections.py
+++ b/Source/time_poincare_sections.py
@@ -0,0 +1,52 @@
 #!/usr/bin/env python
 """
 Time: Poincaré Sections (Linear and Parallel)
 Computes the running time between the linear and parallel algorithms.
@ Author: Moussouni, Yaël (MSc student) & Bhat, Junaid Ramzan (MSc student)
@ Institution:  Université de Strasbourg, CNRS, Observatoire astronomique
                de Strasbourg, UMR 7550, F-67000 Strasbourg, France
@ Date: 2024-11-29
 """
 import time
 import numpy as np
 import main_poincare_sections_linear as lin
 import main_poincare_sections_parallel as par
 E_all = np.array([1/100, 1/12, 1/10, 1/8, 1/6])
 par_time = []
 lin_time = []
 print("\033[34m" + "Please wait..." + "\033[0m")
 for E in E_all:
    t_0 = time.time()
    par.compute_poincare_sections_numpy(E)
    t_1 = time.time()
    par_time.append(t_1-t_0)
 print("\033[34m" + "Still wait..." + "\033[0m")
 for E in E_all:
    t_0 = time.time()
    lin.compute_poincare_sections_linear(E)
    t_1 = time.time()
    lin_time.append(t_1-t_0)
 print("\033[34m" + "Done!" + "\033[0m")
 print("\033[36m" + "=== [ RESULTS ] ===" + "\033[0m")
 print("\033[36m" 
      + "- Linear algorithm:   "
      + "\033[0m"
      + "({:07.4f} ± {:.4f}) s".format(np.mean(lin_time), np.std(lin_time))
      + "\033[36m"
      + " per energy iteration"
      + "\033[0m")
 print("\033[36m" 
      + "- Parallel algorithm: "
      + "\033[0m"
      + "({:07.4f} ± {:.4f}) s".format(np.mean(par_time), np.std(lin_time))
      + "\033[36m"
      + " per energy iteration"
      + "\033[0m")
--- a/activate.sh
+++ b/activate.sh
@@ -0,0 +1,2 @@
 #!/bin/bash
 source ./venv/bin/activate
--- a/area.pdf
+++ b/area.pdf
--- a/mu.pdf
+++ b/mu.pdf
--- a/requirements.txt
+++ b/requirements.txt
@@ -0,0 +1,12 @@
 contourpy==1.3.1
 cycler==0.12.1
 fonttools==4.55.0
 kiwisolver==1.4.7
 matplotlib==3.9.2
 numpy==2.1.3
 packaging==24.2
 pillow==11.0.0
 pyparsing==3.2.0
 python-dateutil==2.9.0.post0
 scipy==1.14.1
 six==1.16.0