CP09_chaos - Examples of nonlinear equations Simple...

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1 1 Nonlinear Differential Equations Nonlinear Differential Equations and The Beauty of Chaos 2 Examples of nonlinear equations ) ( ) ( 2 t kx dt t x d m = Simple harmonic oscillator (linear ODE) More complicated motion (nonlinear ODE) )) ( 1 )( ( ) ( 2 t x t kx dt t x d m α = Other examples: weather patters, the turbulent motion of fluids Most natural phenomena are essentially nonlinear. 3 What is special about nonlinear ODE? Ö For solving nonlinear ODE we can use the same methods we use for solving linear differential equations Ö What is the difference? Ö Solutions of nonlinear ODE may be simple, complicated, or chaotic Ö Nonlinear ODE is a tool to study nonlinear dynamic: chaos, fractals, solitons, attractors A simple pendulum Model: 3 forces • gravitational force • frictional force is proportional to velocity • periodic external force ) cos( ,. ), sin( 2 2 t F dt d mgL dt d I ext f g ext f g ω τ θ β = = = + + = 5 Equations 2 2 2 0 2 0 2 2 , , ) cos( ) sin( mL F f mL L g I mgL t f dt d dt d = = = = + = Computer simulation : there are very many web sites with Java animation for the with Java animation for the simple pendulum 6 Case 1: A very simple pendulum ) sin( 2 0 2 2 = dt d code
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2 7 0 1 02 03 04 05 06 0 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 θ θ (t) time 0.8 8 ) sin( 2 0 2 2 ω = dt d 2 0 2 2 = dt d Is there any difference between the nonlinear pendulum and the linear pendulum? 9 0 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 θ θ (0)=0.2 time 0.8 10 Amplitude dependence of frequency ± For small oscillations the solution for the nonlinear pendulum is periodic with ± For large oscillations the solution is still periodic but with frequency ± explanation: L g = = 0 ϑ < + ) sin( 2 1 ) sin( 2 K L g = < 0 11 Phase-Space Plot velocity versus position -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 θ θ (0)=0.2 d /dt θ E 1 E 2 phase-space plot is a very good way to explore the dynamic of oscillations 0.8 12 Case 2: The pendulum with dissipation dt d dt d α = ) sin( 2 0 2 2 code 0 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 θ (0)=1.0, α =0.1 time How about frequency in this case?
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3 13 Phase-space plot for the pendulum with dissipation 0 1 02 03 04 05 0 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 θ (0)=1.0, α =0.1 θ (t) time -0.5 0.0 0.5 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 d /dt θ 14 Case 3: Resonance and beats ) cos( ) sin( 2 0 2 2 t f dt d ω + = code When the magnitude of the force is very large – the system is overwhelmed by the driven force ( mode locking ) and the are no beats
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CP09_chaos - Examples of nonlinear equations Simple...

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