Complexity and Chaos in Dynamical Systems

Lecturer: Yulia Timofeeva (Office: D2.12, Centre for Complexity Science, Zeeman building)

Lectures: Mon and Tue at 10am-12noon in D1.07

Tutorials: Mon and Tue at 2 - 4pm in D1.07

Lecture Notes

• Introduction, first-order systems
  
• Bifurcations, flows on the circle
  
• Linear systems in R² (in Rⁿ)
  
• Nonlinear systems in R² (in Rⁿ)
  
• Nonlinear oscillations, Hopf bifurcation
  
• More on nonlinear oscillators (Poincare-Bendixson theorem, relaxation oscillators, coupled oscillators, Poincare maps)
  
• One-dimensional maps, logistic map (Slides)
• Lorenz equations (Slides)
  
• Routes to chaos, Fractals (Slides)
• Local and global bifurcations, state-space reconstraction (Slides)
  

Assignments

Computational Tools for Dynamical Systems

• XPPAUT

  XPPAUT - a tool for simulating, animating and analysing dynamical systems

  Installing on Windows (if you already have Xming (an X-server) on your machine, but XPPAUT fails to run, try to additionally install Xming-fonts)
  

  B. Ermentrout (2002) "Simulating, Analyzing, and Animating Dynamical Systems (A guide to XPPAUT for researchers and students)" (SIAM)

  Examples from the book (.ode files)

  Tutorial 1

  Chapter 2 from Ermentrout's book
  

  Chapter 7 from Ermentrout's book
  

  Models
  

  • linear2d.ode
    
  • cusp.ode
    
  • sn.ode
    
  • FitzHugh-Nagumo model
    
  • Relaxation oscillator
    
  • Morris-Lecar model
    
  • Morris-Lecar model (nondimensionalised, different parameters)
    
  • Hodgkin-Huxley model (neuroscience)
    
  • Reduced HH model (neuroscience)
    
  • Two-pool model (calcium signalling)
    
  • De Young Keizer model (calcium signalling)
    
  
  

• MATCONT

  MATCONT - a toolbox for Matlab (more about it)
  

  matcont5p3 download

  Manual (see section 1.3 for setup)
  

  Tutorial 1 Numerical integration

  Tutorial 2 One-parameter bifurcation analysis of steady-states

  Tutorial 3 One-parameter bifurcation analysis of limit cycles
  

  Tutorial 4 Two-parameter bifurcation analysis
  

  
  

• ODE solver for Matlab

  (2D systems, phase-plane analysis, nullclines, trajectories for fixed parameters)