Nonlinear Dynamics

203-2-4721

Course information

Credit points
3.50
Lecture hours
3.00
TA hours
1.00
Lab hours
0.00
University's course list
2024 / 1 2022 / 1 2021 / 1

Summary

Basic theorems in system dynamics (Poincar?-Lyapounov, Hartmann-Grobmann, Center Manifold, KAM); concepts in dynamical systems stability; phase space; Poincar? map; perturbation theory; secular terms; resonance in perturbation theory; method of averaging; method of multiple time scales, method of normal forms; Gronwall lemma; error estimation in approximation methods; freedom i choice of zero-order approximation; forced oscillations; limit cycles; nonlinear diffusion; amplitude equations; nonlinear wave equations; Lorentz equations; chaos in discrete maps-logistic equation; Chaos in Hamiltonian systems; Chaos in dissipative systems; attractors; fractals - basic concepts.

Sources:

Arnold, V. I., Geometrical methods in the Theory of Differential equations (Springer-Verlag, NY, 1988).
Guckenheimer, J. & Holmes, P. J., Nonlinear Oscillations of Vector Fields (Springer-Verlag, NY, 1998).
Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer-Verlag, NY, 1992).
Lichtenberg, A. J. & Lieberman, M. A., Regular and Chaotic Dynamics (Springer-Verlag, NY, 1992).
Abarbanel, H. D. I., Rabinovich, M. I. & Suschnik, M. M., Introduction to Nonlinear Dynamics for Physicicsts (World Scientific, Singapore, 1993).
Hao Bai-Lin, Chaos (World Scientidic, Singapore, 1984).
Kahn, P. B., Mathematical Methods for Scientists & Engineers (Wiley, NY, 1990).
Kahn, P. B. & Zarmi Y., Nonlinear Dynamics – Exploration Through Normal Forms (Wiley, NY, 1998).

Syllabus

  • Surprises in the nonlinear world (Some history and Examples: Logistic equation, Catalysis, Fermi-Pasta-Ulam Problem, Chaotic systems);
  • Concepts in dynamical systems (phase space, fixed points, stability, Poincare map);
  • Basic theorems in system dynamics (Poincare-Lyapounov, Hartmann-Grobmann, Center Manifold, Review of KAM Theorem);
  • Perturbation theory (secular terms, resonance in perturbation theory, Gronwall lemma, error estimation in approximation methods);
  • Perturbation method (method of multiple time scales, method of normal forms, freedom in choice of zero-order approximation);
  • Applications in ODE's (Duffing oscillator, forced oscillations, limit cycles; Lorentz equations);
  • Applications in PDE's (nonlinear diffusion; amplitude equations; nonlinear wave equations - Burgers, KdV & NLS equations and their wave solutions, solitons, compactons);
  • Chaos in discrete maps - logistic equation;
  • Fractals - basic concepts.

Bibliography

Arnold, V. I., Geometrical methods in the Theory of Differential equations (Springer-Verlag, NY, 1988).
Guckenheimer, J. & Holmes, P. J., Nonlinear Oscillations of Vector Fields (Springer-Verlag, NY, 1998).
Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer-Verlag, NY, 1992).
Lichtenberg, A. J. & Lieberman, M. A., Regular and Chaotic Dynamics (Springer-Verlag, NY, 1992).
Abarbanel, H. D. I., Rabinovich, M. I. & Suschnik, M. M., Introduction to Nonlinear Dynamics for Physicicsts (World Scientific, Singapore, 1993).
Nayfeh, A., Perturbation Methods (Wiley, 1978).
Hao Bai-Lin, Chaos (World Scientidic, Singapore, 1984).
Kahn, P. B., Mathematical Methods for Scientists & Engineers (Wiley, NY, 1990).
Kahn, P. B. & Zarmi Y., Nonlinear Dynamics – Exploration Through Normal Forms (Wiley, NY, 1998).

דינמיקה לא לינארית

203-2-4721

נתוני קורס

נקודות זכות
3.50
שעות הרצאה
3.00
שעות תרגול
1.00
שעות מעבדה
0.00
לקובץ הקורסים
2024 / 1 2022 / 1 2021 / 1

תקציר

משפטים בסיסיים בדינאמיקה של מערכות (פואנקרה-ליאפונוב, הרטמן-גרובמן, יריעת המרכז, KAM), מושגים ביציבות של מערכות דינאמיות, מרחב הפאזות, העתקת פואנקרה, תורת ההפרעות, איברים סקולאריים, רזונאנסים בתורת ההפרעות, שיטת המיצוע, שיטת סקאלות הזמן המרובות, תבניות נורמאליות, הלמה של גרונוואל, הערכת השגיאה בשיטות קירוב, החופש בבחירת קירוב האפס בתורת ההפרעות, תנודות מאולצות, מחזור גבולי, דיפוזיה לא-לינארית, משואות אמפליטודה, משואות גלים לא-לינאריות, משואת לורנץ, כאוס בהעתקות דיסקרטיות, כאוס במערכות המילטוניות, כאוס במערכות דיסיפאטיביות, אטרקטורים, מושגים בסיסיים בפרקטאלים.

סילבוס

  • Surprises in the nonlinear world (Some history and Examples: Logistic equation, Catalysis, Fermi-Pasta-Ulam Problem, Chaotic systems);
  • Concepts in dynamical systems (phase space, fixed points, stability, Poincare map);
  • Basic theorems in system dynamics (Poincare-Lyapounov, Hartmann-Grobmann, Center Manifold, Review of KAM Theorem);
  • Perturbation theory (secular terms, resonance in perturbation theory, Gronwall lemma, error estimation in approximation methods);
  • Perturbation method (method of multiple time scales, method of normal forms, freedom in choice of zero-order approximation);
  • Applications in ODE's (Duffing oscillator, forced oscillations, limit cycles; Lorentz equations);
  • Applications in PDE's (nonlinear diffusion; amplitude equations; nonlinear wave equations - Burgers, KdV & NLS equations and their wave solutions, solitons, compactons);
  • Chaos in discrete maps - logistic equation;
  • Fractals - basic concepts.

ביבליוגרפיה

Arnold, V. I., Geometrical methods in the Theory of Differential equations (Springer-Verlag, NY, 1988).
Guckenheimer, J. & Holmes, P. J., Nonlinear Oscillations of Vector Fields (Springer-Verlag, NY, 1998).
Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer-Verlag, NY, 1992).
Lichtenberg, A. J. & Lieberman, M. A., Regular and Chaotic Dynamics (Springer-Verlag, NY, 1992).
Abarbanel, H. D. I., Rabinovich, M. I. & Suschnik, M. M., Introduction to Nonlinear Dynamics for Physicicsts (World Scientific, Singapore, 1993).
Nayfeh, A., Perturbation Methods (Wiley, 1978).
Hao Bai-Lin, Chaos (World Scientidic, Singapore, 1984).
Kahn, P. B., Mathematical Methods for Scientists & Engineers (Wiley, NY, 1990).
Kahn, P. B. & Zarmi Y., Nonlinear Dynamics – Exploration Through Normal Forms (Wiley, NY, 1998).
Last changed on April 25, 2022 by Bar Lev, Yevgeny (ybarlev)