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1MA444 Homepage
A collection of links to various Dynamical Systems resources
Maple worksheets
Timetable
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Course Info:
Course: Applied Dynamical Systems 1MA444
Lecturer: Denis Gaidashev, Ång 14231, gaidash at math.uu.se
First Day Handout
Textbooks:
Primary text:
Nonlinear Dynamics And Chaos: With Applications To Physics,
Biology, Chemistry, And Engineering (Studies in nonlinearity), Steven H.
Strogatz, Publisher: Westview Press.
Secondary text:
Differential Equations, Dynamical Systems, and an Introduction to Chaos , Morris W. Hirsch, Stephen Smale, Robert L. Devaney
Academic Press.
Things on the current agenda:
HOMEWORK 2, DUE 3/12:
Homework 2
PROJECTS:
- Synchronization in chaos (9.6.1, 9.6.2, 9.6.3 in Strogatz)
- Strange attractor in Rossler system (12.3.1, 12.3.2, 12.3.3 in Strogatz)
- Quasiperiodic motion (8.6.2, 8.6.3, 8.6.5, 8.6.6 in Strogatz)
- Correlation dimension of the Lorenz attractor (11.5.1 in Strogatz)
- Method of averaging (7.6.25, 7.6.26 in Strogatz)
- Chemical reaction networks (Chapter 2 in Mathematical Modelling in Systems Biology: An Introduction
by Brian Ingalls, et al. Exercise 2.4.7 )
- Metabolic networks ( Chapter 5 in Mathematical Modelling in Systems Biology: An Introduction by Brian Ingalls, et al. Exercise 5.6.2 )
Example of a program that finds an attracting cycle in the quadratic family, in SageMath, here
LORENZ SYSTEM
An applet, and some applete in the wiki page
The Lorinz attractor folding onto itself
CONSERVATIVE MAPS, HORSESHOES AND CHAOS
Various applets here
My notes about hyperbolic sets and the horseshoe
QUADRATIC MAP
An exposition about the dynamics of the quadratic map
Iterations and the bifurcation diagram for the quadratic map
Another iteration engine for the quadratic map
Intermitency in the quadratic map
Period doubling cascade
Interactive cobweb plot for the quadratic map by Andre Burbanks
CHAOS
Double pendulum applet
Equations of motion for a double pendulumt
Paper by J. Banks et al, "On Devaney's Difinition of Chaos".
A proof of full Sharkovsky theorem, a paper in which Sharkovsky's theorem was rediscovered.
FRACTALS
Serpinski Gasket.
An IFS applet.
IFS,explanation.
Fractal image compression.
A Barnsley fern, explanation.
A Barnsley fern, applet.
Some fractal applets here and many more here
Box counting dimension of coastline, applet
Covering coastline of Norway, applet
Examples of fractals in nature here and here
Devil's staircase for the Cantor set here , here and here
Course Objectives/Outcomes:
1) Understand several fundamental concepts of the dynamical systems, specifically:
- dynamics of the quadratic family,
- topological dynamics; recurrence, mixing and transitivity,
- stable and unstable manifolds, homoclinic and heteroclinic intersections, horseshoes,
- fractals,
- one and two dimensional flows, phase space, limit cycles and Poincar\'e-Bendixson Theorem,
- bifurcations in flows and maps.
2) Understand and be able to explain/present some applications of the theory.
3) Outline the mathematical methods and techniques used to analyse these models and understand in what situations these methods can be applied.
4) Solve representative problems in the above-mentioned topics.
5) Carry out numerical studies of dynamical systems.
Grading:
Two homeworks, 66%. A project 34%
Schedule
- 2019-09-03
- Introduction, course information.
- What is "dynamics"?
- Continous and descrete dynamical systems.
- The quadrtic family as the fundamental example of a chaotic dynamical system.
Reading: Ch 10 in Strogatz
- 2019-09-04
- The quadratic family continued. Periodic and stochastic behaviour.
- Functions vs maps.
- Periodic orbits and their stability.
Reading: Ch 10
- 2019-09-11
- The bifurcation cascade.
- Intermittency and period 3.
- What is chaos?
- Fractals
- Box counting dimension
Reading: Ch 10 and p 323, Ch 11
- 2019-09-16
- Fractals.
- Cardinality and Cantor set.
Reading: Ch 11
- 2019-09-19
- Iterated Function System.
- Hausdorff dimension.
Reading: Ch 11
- 2019-09-26
- Introduction to ODE's.
- One-dimensional flows.
- Bifurcation theory.
Reading: Ch 3
- 2019-09-30
- Bifurcation theory.
- Stable/unstable manifold theorem.
- Pendulum
Reading: Ch 5. Ch 6.0-6.4. Ch 3,4 in Hirsch, Smale, Devaney, Ch 6.7
- 2019-10-04
- Stability of equilibria. Lyapunov function.
- Lyapunov function for damped pendulum and Lorenz system.
Reading: Ch 7.2. Ch 9.2 in Hirsch, Smale, Devaney, Ch 7.5
- 2019-10-07
- Limit sets and cycles.
- Poincare-Bendixson theorem
Reading: Ch 7.
- 2019-10-10
- Poincare-Bendixson theorem
- Oscilating reactions, Selkov system and Hopf bifurcations
- 2D bifurcations.
Reading: Ch 7 (esp 7.3), Ch 8.2, 8.3. Read about bifurcations here
- 2019-10-15
- More about the Selkov system
Pendulum
Reading: 7.3, 6.5
- 2019-10-17
Reading: 6.5, 6.7
- 2019-10-21
- Lorenz equations and chaos
Reading: Chapter 9
- 2019-10-22
- Lorenz equations and chaos
Reading: Chapter 9
- 2019-11-04
- Lorenz equations and chaos
Reading: Chapter 9
- 2019-11-15
- Henon map and horseshoe
- Horseshoe as a fundamental example of chaos
Reading: Chapter 12, exercise 12.1.7
- 2019-11-19
- Henon map and strange attractors
Reading: Chapter 12
- 2019-11-21
- Homoclinic bifurcations, homoclinic tangle.
- Rossler attractor
Reading: 12.0-12.3. A mostly historic review of the discovery of homoclinic penomena by H. Poincare here
- 2019-11-25
- Standard map,
- Tokomak.
- Siegel linearization theorem
Reading: 8.6, also, a much more detailed but accessible proof of the Siegel theorem (simplest KAM theorem) can be found in L. Carleson, Th. Gamelin "Complex Dynamics" in Chapter "Irrationally neutral fixed points".
- 2019-11-27
- Quasiperiodic motion, KAM theory and the standard map
- N-body problem
- Kepler two-body problem
Reading: 8.6, here, here
- 2019-12-03
- N-body problem
- Euler and Lagranfe solutions
- Restricted three body problem
Reading: here
- 2019-12-4
- Michaelis-Menten kinetics;
Reading: here and a presentation here
- 2019-12-10
- Biological oscillators and other models
Reading: here
- 2019-12-12
- Models of enzyme kinetics
Reading:
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