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Course Co-ordinated by IIT Madras
Coordinators
 
Prof. Gaurav Raina
IIT Madras

 

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This course is an introduction to nonlinear dynamics. We will employ a combination of analytical methods, examples, and geometric motivation. We will start with first order differential equations, and their bifurcations, and then move onto two dimensional flows.

A brief introduction to modelling:          
Introduction to the course:         
One dimensional flows:             
     
Flows on the line 

  • Geometrical intuition 
  • Fixed points, and stability       
  • Local stability analysis        
  • Existence and Uniqueness
  • Impossibility of Oscillations
  • Potentials
  • Numerical methods
Bifurcations  
  • Saddle node
  • Normal forms     
  • Transcritical    
  • Pitch fork     
  • Imperfect bifurcations  
  • Example
  • Numerics (XPP-Auto) 
Flows on the circle 
  •         Uniform oscillator
  •         Non-uniform oscillator
Two dimensional flows:    
    
Linear systems  
  •         Definitions and examples
  •         Classification of linear systems
  •         Dynamics of love affairs
Phase plane 
      
  • Phase portraits
  • Existence, uniqueness and topological consequences
  • Fixed points and linearisation
  • Examples: population dynamics
   Limit cycles 
  •         Ruling out closed orbits
  •         Poincare-Bendixson theorem
  •         Lienard systems
  •         Weakly nonlinear oscillators

Bifurcations  
  • Saddle-node, Transcritical and Pitchfork
  • Choice of bifurcation parameter
  • Hopf  
  • Poincare Maps

Final comments:              
  •     Higher-order systems, and the existence of Chaos
  •     Importance of time delays and noise  

Calculus and basic linear algebra


  1. Nonlinear dynamics and Chaos, by Steven Strogatz
  2. Simulating, Analyzing and Animating Dynamical Systems, by Bard Ermentrout


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