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CLASSICA DYNAMIC OF PARTICLES AND SYSTEMS Chapter 4. Nonlinear Oscillations and Chaos. Associate Professor: C. H.L IAO. Contents:. 4.1 Introduction 144 4.2 Nonlinear Oscillations 146 4.3 Phase Diagrams for Nonlinear Systems 150 4.4 Plane Pendulum 155
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CLASSICA DYNAMIC OF PARTICLESAND SYSTEMSChapter 4. Nonlinear Oscillations and Chaos Associate Professor: C. H.LIAO
Contents: • 4.1 Introduction 144 • 4.2 Nonlinear Oscillations 146 • 4.3 Phase Diagrams for Nonlinear Systems 150 • 4.4 Plane Pendulum 155 • 4.5 Jumps, Hysteresis, and Phase Lags 160 • 4.6 Chaos in a Pendulum 163 • 4.7 Mapping 169 • 4.8 Chaos Identification 174
4.1 Introduction • When pressed to divulge greater detail, however, nature insists of being nonlinear; examples are the flapping of a flag in the wind, the dripping of a leaky water faucet, and the oscillations of a double pendulum. • The famous French mathematician Pierre Simon de Laplace espoused the view that if we knew the position and velocities of all the particles in the universe, then we would know the future for all time. • Much of nature seems to be chaotic. In this case, we refer to deterministic chaos, as opposed to randomness, to be the motion of a system whose time evolution has a sensitive dependence on initial conditions.
Ex. 4-1 Sol.:
However, if it had been necessary to stretch each spring a distance d to attach it to the mass when at the equilibrium position, then we would find for the force
4.3 Phase Diagrams for Nonlinear Systems However, in many cases it is difficult to obtain U(x) , and we mustresort to approximation procedures to eventually produce the phase diagram.
By referring to the phase paths for the potentials shown in Figures 4-5 and4-6, we can rapidly construct a phase diagram for any arbitrary potential
let a = 1 and ωo = 1 with appropriate units
FIGURE 4-8 Similar calculation to Figure 4-7 for the solution of the van der PolEquation 4.20. In this case the damping parameter μ = 0.5. Note thatthe solution reaches the limit cycle (now skewed) much more quickly.
4.4 Plane Pendulum FIGURE 4-9 The plane pendulum where the mass m is not required to oscillate insmall angles. The angle θ > 0 is in the counterclockwise direction so that θ0 < 0.
FIGURE 4-10 The component of the force, F(θ), and its associated potential thatacts on the plane pendulum. Notice that the force is nonlinear.
Poincare Section FIGURE 4-19 The damped and driven pendulum for various values of the driving force strength. The angular velocity versus time is shown on the left,and phase diagrams are in the center. Poincare sections are shown onthe right. Note that motion is chaotic for the driving force Fvalues of0.6, 0.7, and l.0.
Problem discussion. Thanks for your attention.
Problem discussion. Problem: • 4-2, 4-4, 4-9, 4-13, 4-17, 4-20, 4-24
#4-20 P.S. : # This is an optional problem.
