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


Asymptotic and qualitative behaviour of ODEs



Playlist Week 1

Introduction, reading list, coursework schedule

Section 1a


1

Introduction to Initial Value Problems, examples

Section 1b


1

Example: the Roessler system

Section 1c


1

Autonomous and non-autonomous ODEs

Section 1d


1

Existence and uniqueness of solutions

Section 1e


1

Flows and solutions, Linear ODEs and the matrix exponential, solution curves

Section 1f


1

Equilibria and limiting behaviour

Section 1g


1

Alpha and omega limit sets

Section 1h


1

Invariant sets

Section 1i


Playlist Week 2

Stability of invariant sets: Liapunov and asymptotic stability

Section 1j


2

Sinks, sources and saddles

Section 1k

Section 1k example Example: direction of asymptotic approach

2

Extra question - flows and limits.

Section 1q1


Section 2 covers approx weeks 2-3

Week

Lectures for Section 2

Link to lecture video

Further viewing

Linear and nonlinear systems


2

Planar systems

Section 2a

Maple plotting phase plane

2

Examples - linear systems in 2d

Section 2b

Section 2x1 [G chapter 5]
Section 2x2

2

Higher dimensional linear systems, and spectra

Section 2c

Playlist Week 3

Unstable, stable and centre subspaces for linear systems

Section 2d

Direct sum of subspaces

3

Nonlinear systems near equilibria

Section 2e

3

Linearization and resonances

Section 2f

Why Jacobians?

3

Further linearization, The Hartman-Grobman theorem 

Section 2g

3

Stable and unstable manifolds for nonlinear systems

Section 2h

Section2x3: Example phase portrait
Manifolds in 2 mins

Section 3 covers approx week 4

Week

Lectures for Section 3

Link to lecture video

Further viewing

Playlist Week 4

Oscillations


4

Periodic orbits

Section 3a

4

Limit cycles, Planar systems and the Poincare index

Section 3b

4

Properties of Poincare index

Section 3c

4

Poincare index of equilibria

Section 3d

4

Example computation of Poincare index

Section 3e

4

Poincare-Bendixson theorem, example

Section 3f (revised 28/10/18)

4

Structural stability

Section 3g

Section 4 covers approx weeks 5-6

Week

Lectures for Section 4

Link to lecture video

Further viewing

Playlist Week 5

Bifurcation theory


5

Introduction to bifurcation theory

Section 4a

5

Continuation and Bifurcations for 1D systems

Section 4b1

Sketching 1D bifucation diagrams using maple

5

Saddle-node bifurcation

Section 4b2

1D bifurcation example (2017, Q1b)

5

Transcritical bifurcation

Section 4c

5

Pitchfork bifurcation

Section 4d

Playlist Week 6

Normal forms for 1D bifurcations

Section 4e

6

Centre manifolds

Section 4f (revised 31/10/19)

6

Centre manifolds with parameters

Section 4g

6

Hopf bifurcation

Section 4h

6

Hopf bifurcation example and further perspectives

Section 4i

Identifying bifurcations


Section 5 covers approx weeks 7, 8 and 9

Week

Lectures for Section 5

Link to lecture video

Further viewing

Playlist Week 7

Chaotic systems


7

From flows to maps

Section 5a

7

Poincare first return map

Section 5b

Poincare section for Roessler

7

Peridic orbits in the plane and hyperbolicity

Section 5c

7

Iterated maps and orbits

Section 5d

Sketching iterated maps

7

Periodic, eventually periodic and aperiodic orbits of iterated maps

Section 5e

7

The sawtooth/doubling map

Section 5f

Binary expansions

7

Properties of the sawtooth/doubling map

Section 5g

Playlist Week 8

Liapunov exponent

Section 5h

Scholarpedia page

8

Horseshoes and chaos

Section 5i

8

Horseshoe lemma

Section 5j

8

Cantor sets

Section 5k

8

Example: logistic map, fixed points and bifurcations

Section 5la

8

Period doubling cascade for logistic map

Section 5lb

8

Period 3 implies chaos (Li and Yorke theorem)

Section 5m

J Yorke talks about chaos

Playlist Week 9

Sharkovsky's theorem

Section 5n

O Sharkovsky giving a talk

9

Admissible graphs and minimal period three

Section 5o

9

Periodic points and admissible sequences

Section 5p

9

Sharkovsky's theorem idea of proof

Section 5q

Scholarpedia page

9

Example: Tent map

Section 5r

9

Example: Times ten map

Section 5s

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