MATH383: Dynamical system linearization, period-doubling, chaotic behavior

Overview

• Local linearization, eigenvalue evolution maps

• Forced systems, period doubling

• Chaotic behavior, implications for nonlinear systems: weather prediction, economic forecasts

Local linearization

• Behavior of solutions of nonlinear system ${𝒚}^{\text{'}}=𝒇\left(𝒚\right),𝒚\in {ℝ}^n$ can be approximated locally by

  ${\displaystyle {𝒚}^{\text{'}}\cong {𝒇}_k+{𝑨}_k(𝒚-{𝒚}_k),{𝒇}_k=𝒇\left({𝒚}_k\right),{𝑨}_k=\frac{\partial 𝒇}{\partial 𝒚}\left({𝒚}_k\right)}$

• In the vicinity of each point ${𝒚}_k$, eigenvalues ${\lambda }_{k,1},\dots ,{\lambda }_{k,n}$ of ${𝑨}_k$ indicate localized behavior:

  • periodic orbits $\mathrm{Re}{\lambda }_{k,j}=0$

  • attractor ${\lambda }_{k,j}<0$ for all $j$

  • repeller ${\lambda }_{k,j}>0$ for all $j$

  • saddle points ${\lambda }_{k,j}\in ℝ$ of differing signs

Forced systems

• Consider now behavior of solutions of forced nonlinear system ${𝒚}^{\text{'}}=𝒇\left(𝒚\right)+𝑭\left(t\right),𝒚\in {ℝ}^n$

• Local linearization is

  ${\displaystyle {𝒚}^{\text{'}}\cong {𝒇}_k+{𝑨}_k(𝒚-{𝒚}_k)+𝑭\left(t\right),{𝒇}_k=𝒇\left({𝒚}_k\right),{𝑨}_k=\frac{\partial 𝒇}{\partial 𝒚}\left({𝒚}_k\right)}$

• The behavior now depends on how each possible behavior is influenced by the force $𝑭\left(t\right)$. As in the unforced case, In the vicinity of each point ${𝒚}_k$, eigenvalues ${\lambda }_{k,1},\dots ,{\lambda }_{k,n}$ of ${𝑨}_k$ indicate localized behavior, but can now be modified by $𝑭\left(t\right)$

  • periodic orbits $\mathrm{Re}{\lambda }_{k,j}=0$, modified by the periods of $𝑭\left(t\right)$: resonance, period doubling

  • attractor ${\lambda }_{k,j}<0$ for all $j$, modified by $𝑭\left(t\right)$: reinforce, counteract dissipation

  • repeller ${\lambda }_{k,j}>0$ for all $j$, modified by $𝑭\left(t\right)$: system stabilization

  • saddle points ${\lambda }_{k,j}\in ℝ$ of differing signs, modified by $𝑭\left(t\right)$: maintain unstable system equilibrium

• See Homework12 solution for numerical experiments showing the above behaviors

Chaotic behavior

• Systems can exhibit complex behavior when periods in the forcing and those intrinsic to the system are incomensurable

• Such behavior is recognized by complex Poincare sections, typically fractal, i.e., of non-integer dimension, and is termed “chaotic”

• Chaotic behavior arises surprisingly often, and is considered to be generic (i.e., expected), in the sense that a forced, nonlinear system will typically show chaotic behavior for some choice of system parameters

• Famous example: Lorenz system for weather prediction

  If the flap of a butterfly's wings can be instrumental in generating a tornado, it can equally well be instrumental in preventing a tornado. (E. Lorenz, 1972)

• Further study: Dynamical systems, Ergodic Theory (MATH590, MATH897)