1

2 Class #26 Nonlinear Systems and Chaos Most important concepts  Sensitive Dependence on Initial conditions  Attractors Other concepts  State-space orbits  Non-linear diff. eq.  Driven oscillations  Second Harmonic Generation  Subharmonics  Period-doubling cascade  Bifurcation plot  Poincare diagram  Feigenbaum number  Universality

3 Outline Origins and Definitions of chaos State Space Behavior of a driven damped pendulum (DDP) Non-linear behavior of a DDP Attractor Period doubling Sensitive dependence Bifurcation Plot

4 Definition of chaos The dynamical evolution that is aperiodic and sensitively dependent on initial conditions. In dissipative dynamical systems this involves trajectories that move on a strange attractor, a fractal subspace of the phase space. This term takes advantage of the colloquial meaning of chaos as random, unpredictable, and disorderly behavior, but the phenomena given the technical name chaos have an intrinsic feature of determinism and some characteristics of order. Colloquial meaning – disorder, randomness, unpredictability Technical meaning – Fundamental unpredictability and apparent randomness from a system that is deterministic. Some real randomness may be included in real systems, but a model of the system without ANY added randomness should display the same behavior.

5 Weather and climate prediction Is important  Can we go: a) For a hike? b) Get married outdoors? c) Start a war?  Will we be hurt by a: a) Tornado? b) Hurricane? c) Lightning bolt?  How much more fossil fuel can we burn before we: a) Fry? b) Drown? c) Starve?

6 Early work by Edward N. Lorenz 1960’s at MIT  Early computer models of the atmosphere Were very simple (Computers were stupid) Were very helpful Results were not reproducible!!  Lorenz ultimately noticed that For 7-10 days of prediction, all of his models reproduced very well. After 7-10 days, the same model could be run twice and give the same result – BUT!! Changes that he thought were trivial (e.g. changing the density of air by rounding it out at the 3 th decimal place, or slightly modifying the initial conditions at beginning of model) Produced COMPLETELY DIFFERENT results  This came to be called “The Butterfly effect”

7 State Space / Phase Space

8 Viscous Drag III – Stokes Law Form-factor k becomes “D” is diameter of sphere Viscous drag on walls of sphere is responsible for retarding force. George Stokes [ ]  (Navier-Stokes equations/ Stokes’ theorem) D

9 Damped Driven Pendulum (DDP) Damping  Pendulum immersed in fluid with Newtonian viscosity  Damping proportional to velocity (and angular velocity) Driving  Constant amplitude drive bar  Connected to pendulum via torsion spring  Torque on Pendulum is

10 Damped Driven Pendulum (DDP)

11 Conditions for chaos Dissipative Chaos 1. Requires a differential equation with 3 or more independent variables. 2. Requires a non-linear coupling between at least two of the variables. 3. Requires a dissipative term (that will use up energy). Non-dissipative chaos Not in this course

12 Sensitive dependence on initial conditions

13 Sensitive dependence on initial conditions

14 Worked problem Sketch a state-space plot for the magnetic pendulum Indicate the attractors and repellers Show some representative trajectories First explore the trajectories beginning with theta- dot=0 Then explore trajectories that begin with theta in some state near an attractor but through proper choice of theta-dot move to the other attractor. Sketch the basins of attraction