Classical Dynamics

A Contemporary Approach

by Jorge V. José and Eugene Saletan
Classical Dynamics Book

Chapter Seven:

Nonlinear Dynamics






Fig. 6.32sm

Fig 6.43S

 
Nonlinear oscillators
  1. 1. a model system
  2. 2. driven quartic oscillator: damped driven quartic oscillator, undamped driven quartic oscillator
  3. 3. example: the van der pol oscillator

Stability of solutions
  1. 1. stability of autonomous systems: definitions, the Poincar&eacute-Bendixon theorem, linearization
  2. 2. stability of non-autonomous systems: the Poincaré map, linearization of discrete maps, example: the linearized H&eacutenon map

Parametric oscillators
  1. 1. Floquet theory: the Floquet operator R, standard basis, eigenvalues of R and stability, dependence on G
  2. 2. the vertically driven pendulum: the Mathieu equation, stability of the pendulum, the inverted pendulum, damping

Discrete maps, chaos
  1. 1. the logistic map: definition, fixed points, period doubling, universality, further remarks
  2. 2. the circle map: the damped driven pendulum, the standard sine circle map, rotation number and the devil's staircase, fixed points of the circle map

Chaos in Hamiltonian systems and the KAM theorem
  1. 1. the kicked rotator: the dynamical system, the standard map, Poincaré map of the perturbed systems
  2. 2. the H&eacutenon map
  3. 3. chaos in Hamiltonian systems
  4. 4. the KAM theorem:background, two-conditions: Hessian and diophantine, the theorem, a brief description of the proof of KAM

Appendix on number theory
  1. 1. the unit interval
  2. 2. a diophantine condition
  3. 3. the circle and the plane
  4. 4. KAM and continued fractions

Problems
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