מערכות דינמיות וכאוס, 2023א

About course - General

• מערכות רבות בעולם החי והדומם מציגות דינמיקה לא לינארית הבאה לידי ביטוי בדרכים שונות: איבוד יציבות של מצב מערכת והתפתחותו למצב אחר תוך שבירת סימטריה, תגובה קיצונית להפרעות קטנות, התנהגויות אוניברסליות ועוד. מערכות אלו מתוארות לרוב על ידי משוואות דיפרנציאליות לא לינאריות רגילות (מד"ר) או חלקיות (מד"ח). הקורס נועד להקנות כלים בסיסיים, אנליטיים וחישוביים, לניתוח מערכות לא לינאריות, תוך הדגמתם בעזרת מודלים של מערכות פיזיקליות, סביבתיות, כימיות וביולוגיות.
  

  
  

  Many living and inanimate systems exhibit nonlinear dynamics expressed in various ways: loss of system state stability and its evolution to another by symmetry breaking, extreme response to small disturbances, universal behaviors and more. These systems are most often described by ordinary (non-linear) or partial (non-linear) differential equations. The course aims to provide basic, analytical and computational tools for analyzing nonlinear systems, demonstrating them with models of physical, environmental, chemical and biological systems.
  

  
  

  זמן סופי של השיעורים יקבע בתיאום עם הסטודנטים הנרשמים

  The final times of the lectures will be determined together with the enrolled students

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    Link to the BGU course listing 203.2.5421 URL
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    לוח הודעות Forum
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    Zoom Links Zoom@BGU

Topic outline

• Topics

  
  

  Week	Topic	Additional reading
  1	Formalities, motivation, and course overview Lecture slides At home, watch 3 movies from slide 33 and onward	Strogatz, "Nonlinear Dynamics and Chaos", Chapter 1 and Sections 2.0-2.3
  2	Linear stability in 1D, bifurcation theory, and numerical methods for ODEs Lecture slides (before the lecture, watch the full movie on slides 1/2) Home exercises - submission before the next class	Strogatz, "Nonlinear Dynamics and Chaos", Sections 3.0-3.2, 3.4, 2.8
  3	Qualitative analysis: effect of time scales and parameters Lecture slides (before the lecture, watch the full movies on slides 1/2 and 7/8) Home exercises - submission before the next class	Strogatz, "Nonlinear Dynamics and Chaos", Sections 3.5,3.7
  4	Linear and nonlinear 2D systems: fixed points and manifolds Lecture slides (before the lecture, watch the full movies on slides 1/2 and 7/8) Home exercises - submission before the next class	Strogatz, "Nonlinear Dynamics and Chaos", Chapter 5 and Sections 6.0-6.4
  5	Conservative systems and limit cycles Lecture slides (before the lecture, watch the movies on slides 1/2 and 14/15) Home exercises - submission before the next class	Strogatz, "Nonlinear Dynamics and Chaos", Sections 6.5-6.7, 7.0-7.1
  6	Closed orbits (Poincaré-Bendixson theorem) and nonlinear oscillations Lecture slides (before the lecture, watch the movies on slides 1/2 and 10/11) Home exercises - submission before the next class	Strogatz, "Nonlinear Dynamics and Chaos", Sections 7.2-7.6 Additional reading from the book of Edelstein-Keshet on Mathematical Models in Biology (Chapter 8)
  7	Generalization of bifurcations in 2D and center manifold reduction Lecture slides (before the lecture, watch the movies on slides 1/2 and 4/5) Home exercises - submission before the next class	Strogatz, "Nonlinear Dynamics and Chaos", Sections 8.0-8.3 Review paper: Crawford, Introduction to bifurcation theory, Rev. Mod. Phys., 1991
  8	Stability (Floquet theory) and global bifurcations of cycles Lecture slides (before the lecture, watch the movies on slides 1/2) Home exercises - submission before class #9	Strogatz, "Nonlinear Dynamics and Chaos", Sections 8.4-8.7 Easy paper about the stability of cycles: Klausmeier, Floquet theory: a useful tool for understanding nonequilibrium dynamics, Theor. Ecol., 2008
  20/12/22	Mid-term project presentations by students	Mid-term project guidelines
  9	Lorentz equations and chaos Lecture slides (before the lecture, watch the movies on slides 1/2 and 4/5 and 7/8) Home exercises - submission before the next class	Strogatz, "Nonlinear Dynamics and Chaos", Sections 9.1-9.3, 12.0-12.1, 12.4 Short notes that include some of the topics: Nonlinear Dynamical Systems, Their Stability, and Chaos Additional reading: Sandri, Numerical calculation of Lyapunov exponents
  10	One dimensional maps (Poincaré sections), period-doubling, and route to chaos  Lecture slides (before the lecture, watch the movies on slides 1/2 and 4/5 and 7/8) Home exercises - submission before the next class	Strogatz, "Nonlinear Dynamics and Chaos", Sections 10.0-10.6 Additional reading: Li and Chen, Estimating the Lyapunov exponents of discrete systems, Chaos, 14, 343 (2004)
  11	Aperiodic dynamics, frequency locking, and Fractals Lecture slides (before the lecture, watch the movies on slides 1/2 and 26/27 and 28/29) Home exercises - submission before the next class	Strogatz, "Nonlinear Dynamics and Chaos", Sections 8.6, 10.7, 11.0-11.3, 12.0-12.3
  12	Introduction to pattern formation (nonlinear PDEs) Lecture slides	Short and popular overview: Whitesides and Grzybowski, Self-assembly at all scales,  Science 295, 2418 (2002) Detailed and extended review: Cross and Hohenberg, Pattern formation outside of equilibrium, Reviews of Modern Physics 65, 851 (1993)
  25/3/23	Final project, folder with articles	guidelines

  
  

• Lecture, Tutorial and office hours

  Office hours

  Name	Day	Hours	Building/Room	E-mail
  Arik Yochelis	TBD	By appointment	54/220	yochelis@bgu.ac.il

  
  

  Lecture/Tutorial

  Group	What?	Name	Day	Hours	Building/Room
  1	Lecture/mandatory	Arik Yochelis	Tuesday	09:00-11:00	room 9, bld. 34

• Textbooks

  Main book

  • S.H. Strogatz, Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry and Engineering, 2^nd Edition, CRC Press 2015

  • YouTube to all lecture movies

  
  

  Additional literature

  • D. Kaplan and L. Glass, Understanding Nonlinear Dynamics, Springer Science+Business Media New York, 1995

  • J.D. Crawford, Introduction to bifurcation theory, Rev. Mod.; Phys., 1991

  • J.D. Murray, Mathematical Biology (Parts I), 3^rd Edition, Springer, 2002

  
  

• Course policy

  Classes: TBD.

  Attendance is mandatory with exceptions upon lecturer approval.

  Grading

  • Homework submission, 10%
  • Mid-term project, 30%
  • Final project, 60%

  General

  • Email: should only be used for communication on personal matters, all questions of relevance to more than one person should be asked via the Q&A forum, to avoid repetition.