Dynamical Systems video lectures
MAGIC020 Dynamical Systems
Section 1 covers approx week 1-2
Week |
Lectures for Section 1 |
Link to lecture video |
Further viewing |
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Asymptotic and qualitative behaviour of ODEs |
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Introduction, reading list, coursework schedule |
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1 |
Introduction to Initial Value Problems, examples |
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1 |
Example: the Roessler system |
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1 |
Autonomous and non-autonomous ODEs |
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1 |
Existence and uniqueness of solutions |
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1 |
Flows and solutions, Linear ODEs and the matrix exponential, solution curves |
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1 |
Equilibria and limiting behaviour |
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1 |
Alpha and omega limit sets |
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1 |
Invariant sets |
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Stability of invariant sets: Liapunov and asymptotic stability |
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2 |
Sinks, sources and saddles |
Section 1k example Example: direction of asymptotic approach |
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2 |
Extra question - flows and limits. |
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Section 2 covers approx weeks 2-3
Week |
Lectures for Section 2 |
Link to lecture video |
Further viewing |
Linear and nonlinear systems |
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2 |
Planar systems |
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2 |
Examples - linear systems in 2d |
Section 2x1
[G chapter 5] |
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2 |
Higher dimensional linear systems, and spectra |
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Unstable, stable and centre subspaces for linear systems |
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3 |
Nonlinear systems near equilibria |
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3 |
Linearization and resonances |
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3 |
Further linearization, The Hartman-Grobman theorem |
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3 |
Stable and unstable manifolds for nonlinear systems |
Section2x3:
Example phase portrait |
Section 3 covers approx week 4
Week |
Lectures for Section 3 |
Link to lecture video |
Further viewing |
Oscillations |
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4 |
Periodic orbits |
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4 |
Limit cycles, Planar systems and the Poincare index |
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4 |
Properties of Poincare index |
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4 |
Poincare index of equilibria |
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4 |
Example computation of Poincare index |
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4 |
Poincare-Bendixson theorem, example |
Section 3f (revised 28/10/18) |
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4 |
Structural stability |
Section 4 covers approx weeks 5-6
Week |
Lectures for Section 4 |
Link to lecture video |
Further viewing |
Bifurcation theory |
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5 |
Introduction to bifurcation theory |
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5 |
Continuation and Bifurcations for 1D systems |
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5 |
Saddle-node bifurcation |
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5 |
Transcritical bifurcation |
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5 |
Pitchfork bifurcation |
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Normal forms for 1D bifurcations |
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6 |
Centre manifolds |
Section 4f (revised 31/10/19) |
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6 |
Centre manifolds with parameters |
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6 |
Hopf bifurcation |
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6 |
Hopf bifurcation example and further perspectives |
Section 5 covers approx weeks
7, 8 and 9
Week |
Lectures for Section 5 |
Link to lecture video |
Further viewing |
Chaotic systems |
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7 |
From flows to maps |
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7 |
Poincare first return map |
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7 |
Peridic orbits in the plane and hyperbolicity |
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7 |
Iterated maps and orbits |
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7 |
Periodic, eventually periodic and aperiodic orbits of iterated maps |
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7 |
The sawtooth/doubling map |
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7 |
Properties of the sawtooth/doubling map |
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Liapunov exponent |
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8 |
Horseshoes and chaos |
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8 |
Horseshoe lemma |
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8 |
Cantor sets |
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8 |
Example: logistic map, fixed points and bifurcations |
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8 |
Period doubling cascade for logistic map |
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8 |
Period 3 implies chaos (Li and Yorke theorem) |
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Sharkovsky's theorem |
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9 |
Admissible graphs and minimal period three |
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9 |
Periodic points and admissible sequences |
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9 |
Sharkovsky's theorem idea of proof |
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9 |
Example: Tent map |
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9 |
Example: Times ten map |
Please let me know if any of the material in the videos is unclear, inaudible, incorrect, needs further clarification etc etc!
Peter Ashwin, Autumn Term 2021-22
P.Ashwin@exeter.ac.uk