Temporal Patterns in Nonlinear Systems

203-2-4931

Course information

Credit points
3.00
Lecture hours
3.00
TA hours
0.00
Lab hours
0.00
University's course list
2024 / 2 2022 / 1 2021 / 1 2020 / 1 2018 / 1 2016 / 1 2014 / 2

Summary

Instabilities in physical systems: Rayleigh-Benard convection, self focusing and optical solitons, chemical oscillations, vegetation patterns. Mathematical analysis of instabilities: transcritical, saddle node, pitchfork and Hopf bifurcations. Primary instabilities in spatially extended systems: applications to convection and chemical oscillations. Amplitude equations: Non-Linear Shrodinger (NLS) equation, Newell-Whitehead-Segel (NWS) equation, Complex Ginzburg-Landau (CGL) equation. Secondary instabilities: Eckhaus, zigzag and Benjamin-Feir instabilities. Phase and amplitude turbulence. Front propagation in bistable systems: gradient vs. non-gradient systems, the non-equilibrium Ising-Bloch (NIB) bifurcation, transverse front instabilities. Singular perturbation theory of stationary and traveling patterns in bistable and excitable media. Spiral waves: kinematic theory, spiral turbulence.


Sources:

S.H. Strogatz, Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry and Engineering, (Addison-Wesley, Reading Mass., 1994). Q 172.5.C45S767.
P. Manneville, Dissipative Structures and Weak Turbulence, (perspectives in Physics, New York, 1990). QA 871 .M33.
A.C. Newell and J.V. Moloney, Nonlinear Optics, (Addison-Wesley, New York, 1992). QC446.2.N48.
M. Cross and P .C. Hohenberg, Pattern Formation outside of Equilibrium, Rev. Mod. Phys. 65 (1993) 2.
E. Meron, Pattern Formation in Excitable Media, Phys. Rep. 218 (1992) 1.

Syllabus

  1. Instabilities in physical systems: Rayleigh-Benard convection. Self focusing and optical solitons. Chemical oscillations. Vegetation patterns.
  2. Mathematical formulation of instabilities: The transcritical, saddle node, pitchfork and Hopf bifurcations. Imperfect bifurcations. Local and global bifurcations. Excitable systems.
  3. Spatio-temporal patterns in small systems. Normal forms and the Center Manifold theorem.
  4. Amplitude equations. Applications to electromagnetic waves in dispersive media, convection, and chemical oscillations. The Non-Linear Shrodinger (NLS) equation, the Newell-Whitehead-Segei (NWS) equation, and the Complex Ginzburg-Landau (CGL) equation.
  5. Phase dynamics. The Eckhaus, zigzag and Benjamin-Feir instabilities. Phase and amplitude turbulence.
  6. Front propagation in bistable systems. Gradient vs. non-gradient systems. The non-equilibrium Ising-Bloch (NIB) bifurcation. Transverse instabilities.
  7. Singular perturbation theory of stationary and traveling patterns in bistable and excitable media.
  8. Spiral waves. Spiral turbulence. Kinematic theory.
  9. Forced oscillations. Frequency locking in spatially extended systems.

Bibliography

S.H. Strogatz, Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry and Engineering, (Addison-Wesley, Reading Mass., 1994). Call #:Q 172.5.C45S767.
M. Cross and H. Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems, (Cambridge University Press, 2009).
P. Manneville, Dissipative Structures and Weak Turbulence, (perspectives in Physics, New York, 1990). Call #: QA 871 .M33.
A.C. Newell and J.V. Moloney, Nonlinear Optics, (Addison-Wesley, New York, 1992). Call #: QC446.2.N48.
J.D. Murray, Mathematical Biology, (Springer-Verlag, New York, 1989). Call #: QH 323.5.M88.
J. Guckenheimer and P. Holmes Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, (springer-Verlag, New York, 1983). Call #: OA 1.A647.
M. Cross and P .C. Hohenberg, Pattern Formation outside of Equilibrium, Rev. Mod. Phys. 65 (1993) 2.
J.J. Tyson and J.P. Keener Singular Perturbation Theory of 8. Traveling Waves in Excitable Media, Physica D 32 (1988) 327.
E. Meron, Pattern Formation in Excitable Media, Phys. Rep. 218 (1992) 1.

תצורות מרחב וזמן במערכות לא לינאריות

203-2-4931

נתוני קורס

נקודות זכות
3.00
שעות הרצאה
3.00
שעות תרגול
0.00
שעות מעבדה
0.00
לקובץ הקורסים
2024 / 2 2022 / 1 2021 / 1 2020 / 1 2018 / 1 2016 / 1 2014 / 2

תקציר

אי יציבויות במערכות פיסיקליות: הסעת Rayleigh Benard, מיקוד עצמי וסוליטונים אופטיים, גלים כימיים, פוטנציאלי פעולה בממברנות אקסיטביליות. תאור מתמטי של אי יציבויות. תצורות מרחב וזמן במערכות קטנות. צורות נורמליות ומשפט היריעה המרכזית (Center Manifold). משוואות אמפליטודה. יישומים לגלים אלקטרומגנטיים בתווך נפיצתי, זרימה ותנודות כימיות. משוואת שרדינגר הלא לינארית (NLS). משוואת ניואל-וויטהד-סגל (NWS). משוואת גינזבורג לנדאו המורכבת (CGL). דינמיקת פאזה. משוואות Kuramoto ו- Pomeau-Manneville. אי יציבויות זיגזג, Eckhaus ו- Benjamin-Feir. טורבולנציות פאזה ואמפליטודה. חזיתות גל בתווך ביסטבילי. מערכות גרדיאנטיות לעומת לא-גרדיאנטיות. מעברים מסוג Bloch-Isingרחוק משיווי משקל. תורת הפרעה סינגולרית של תצורות נייחות ונעות בתווך ביסטבילי ואקסיטבילי. גלי ספירלה. טורבולנצית מערבולות. בקרת תצורות.

סילבוס

  1. Instabilities in physical systems: Rayleigh-Benard convection. Self focusing and optical solitons. Chemical oscillations. Vegetation patterns.
  2. Mathematical formulation of instabilities: The transcritical, saddle node, pitchfork and Hopf bifurcations. Imperfect bifurcations. Local and global bifurcations. Excitable systems.
  3. Spatio-temporal patterns in small systems. Normal forms and the Center Manifold theorem.
  4. Amplitude equations. Applications to electromagnetic waves in dispersive media, convection, and chemical oscillations. The Non-Linear Shrodinger (NLS) equation, the Newell-Whitehead-Segei (NWS) equation, and the Complex Ginzburg-Landau (CGL) equation.
  5. Phase dynamics. The Eckhaus, zigzag and Benjamin-Feir instabilities. Phase and amplitude turbulence.
  6. Front propagation in bistable systems. Gradient vs. non-gradient systems. The non-equilibrium Ising-Bloch (NIB) bifurcation. Transverse instabilities.
  7. Singular perturbation theory of stationary and traveling patterns in bistable and excitable media.
  8. Spiral waves. Spiral turbulence. Kinematic theory.
  9. Forced oscillations. Frequency locking in spatially extended systems.

ביבליוגרפיה

S.H. Strogatz, Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry and Engineering, (Addison-Wesley, Reading Mass., 1994). Call #:Q 172.5.C45S767.
M. Cross and H. Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems, (Cambridge University Press, 2009).
P. Manneville, Dissipative Structures and Weak Turbulence, (perspectives in Physics, New York, 1990). Call #: QA 871 .M33.
A.C. Newell and J.V. Moloney, Nonlinear Optics, (Addison-Wesley, New York, 1992). Call #: QC446.2.N48.
J.D. Murray, Mathematical Biology, (Springer-Verlag, New York, 1989). Call #: QH 323.5.M88.
J. Guckenheimer and P. Holmes Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, (springer-Verlag, New York, 1983). Call #: OA 1.A647.
M. Cross and P .C. Hohenberg, Pattern Formation outside of Equilibrium, Rev. Mod. Phys. 65 (1993) 2.
J.J. Tyson and J.P. Keener Singular Perturbation Theory of 8. Traveling Waves in Excitable Media, Physica D 32 (1988) 327.
E. Meron, Pattern Formation in Excitable Media, Phys. Rep. 218 (1992) 1.
Last changed on April 25, 2022 by Bar Lev, Yevgeny (ybarlev)