Oxford notes

Discrete-time dynamical systems are sometimes called maps. As usual, there are linear maps, which cam be represented by a matrix (plus a constant vector, if the map is affine, instead of just linear). However, most interesting behaviour is observed in nonlinear maps, in which the state at discrete time $n+1$ depends on the state at the previous time via a nonlinear function $f$:

$x_{n+1} = f(x_n ; n)$

where we allow discrete-time dependence of $f$. Autonomous maps, won't have such dependence.

Poincare maps

Cross-sections of the phase plane of a Continuous dynamical system that are nowhere tangential to a trajectory are called Poincare sections. Trajectories become points in the lower dimensional space of the cross-section, and the dynamical system becomes a discrete map, called the Poincare map.

Features of maps

The equivalent of equilibrium points in dynamical systems are fixed points. A fixed point is one that is mapped to itself.

Periodic cycles are closed orbits (like limit cycles, or orbits, in dynamical systems).

Stability

The stability of a fixed a fixed point is determined by its multiplier (which is just the derivative of the function defining the map $\lambda = \frac{\partial f}{\partial x}$) evaluated at the fixed point. A point is stable if $|\lambda| <1$, unstable if $|\lambda| >1$, and neutrally stable if $|\lambda| =1$ (at which point a bifurcation occurs).

One can use the Jury test to find if the roots of a polynomial are inside the unit circle, which is useful for stability.

The stability of a periodic cycle can be found by multiplying the multiplier evaluated at each of the points in the cycle. These numbers are then called the characteristic multipliers or Floquet exponents.

Bifurcations in 1D maps

• Fold bifurcations. Bifurcations that occur when the multiplier $\lambda =1$. Can be:
  • Saddle-node
  • Transcritical
  • Pitchfork
• Flip or period-doubling bifurcation. Occur when the multiplier $\lambda = -1$
• Hopf bifurcation. Occur when the multiplier $\lambda = e^{i\theta}$ (for a $\theta \neq 0, \pi$, I suppose).

One can also have bifurcations of periodic cycles in 2D maps, I think.

There are also global bifurcations in periodic maps, some of which are routes that lead to chaos. See Nonlinear dynamical systems and Chaos theory.

2D maps

Local linear stability analysis done by Jacobians, and multipliers are replaced by the Jacobian's eigenvalues which must now be less than one in magnitude for stability. For periodic cycles, one multiples the Jacobians.

Another very interesting feature of nonlinear maps, is that many of them exhibit chaos.

Examples

Henon map

Standard (or Chirikov) map

See more examples of chaotic maps in Chaos theory