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12 Nonlinear Mechanics and Chaos Jen-Hao Yeh Linear Nonlinear easy hard Uncommon common analytical numerical Superposition principle Chaos Nonlinear Chaotic Linear Driven, Damped Pendulum (DDP) 2 2 2 sin( ) cos(t ) 0 2 b / m 02 g / L 0 F0 / mg We expect something interesting to happen as → 1, i.e. the driving force becomes comparable to the weight A Route to Chaos NDSolve in Mathematica Fig12p2 NDSolve '' t 2 InterpolatingFunction Plot Evaluate t ' t 0 ^2 0., 6. Sin t 0 ^2 Cos t , 0 0, ' 0 , . Fig12p2 , t, 0, 6 , Ticks 1, 2, 3, 4, 5, 6 , 0.3, 0.3 0.3 1 2 3 4 0 , , t, 0, 6 5 6 0.3 Download Mathematica for free: https://terpware.umd.edu/Windows/Package/2032 , PlotRange All, AxesS Driven, Damped Pendulum (DDP) 2 2 2 sin( ) cos(t ) 0 For all following plots: 2 Period = 2/ = 1 0 1.5 0 / 4 (0) (0) 0 0 Small Oscillations of the Driven, Damped Pendulum << 1 will give small oscillations 2 2 0 0 2 cos(t ) = 0.2 (t) 0.3 1 2 3 4 5 6 t 0.3 Calculated with Mathematica NDSolve After the initial transient, the solution looks like (t ) A cos(t ) Periodic “attractor” Fig. 12.2 Small Oscillations of the Driven, Damped Pendulum << 1 will give small oscillations 1) The motion approaches a unique periodic attractor independent of initial conditions 2) The motion is sinusoidal with the same frequency as the drive (t ) A cos(t ) Moderate Driving Force Oscillations of the Driven, Damped Pendulum < 1 and the nonlinearity becomes significant… 1 3 2 2 2 0 0 cos(t ) 6 Try (t ) A cos(t ) This solution gives from the 3 3 term: cos x 1 cos 3x 3 cos x 4 Since there is no cos(3t) on the RHS, it must be that , , all develop a cos(3t) time dependence. Hence we expect: (t ) A cos(t ) B cos3(t ) B A We expect to see a third harmonic as the driving force grows Harmonics Stronger Driving: The Nonlinearity Distorts the cos(t) (t) (t) = 0.9 cos(t) 2 2 1 2 3 4 5 6 t t 5 6 -cos(3t) 2 2 The motion is periodic, but … The third harmonic distorts the simple (t ) A cos(t ) Fig. 12.3 Even Stronger Driving: Complicated Transients – then Periodic! After a wild initial transient, the motion becomes periodic (t) = 1.06 4 2 t 3 6 9 12 15 After careful analysis of the long-term motion, it is found to be periodic with the same period as the driving force Fig. 12.4 Slightly Stronger Driving: Period Doubling After a wilder initial transient, the motion becomes periodic, but period 2! (t) 4 = 1.073 2 t 5 2 10 15 20 25 8. 30 8.5 22 24 26 28 30 The long-term motion is TWICE the period of the driving force! A SUB-Harmonic has appeared Fig. 12.5 Harmonics Sub-harmonic Slightly Stronger Driving: Period 3 The period-2 behavior still has a strong period-1 component Increase the driving force slightly and we have a very strong period-3 component = 1.077 (t) 4 2 Period 3 2 4 6 8 10 12 14 t Fig. 12.6 Multiple Attractors The linear oscillator has a single attractor for a given set of initial conditions For the driven damped pendulum: Different initial conditions result in different long-term behavior (attractors) = 1.077 (t) (0) 0, (0) 0 4 Period 3 2 (0) / 2, (0) 0 2 4 6 8 10 12 14 t Period 2 Fig. 12.7 Period Doubling Cascade (t) = 1.06 (t) t 2 4 6 8 10 2 = 1.078 (0) 0 2 Close-up of steady-state motion Period 1 2.5 2 30 32 34 36 38 40 30 Period 2 32 34 36 38 40 t 2 2 4 6 8 10 2 2.5 2 2 = 1.081 2 4 6 8 10 2 2.5 2 30 32 Period 4 34 36 38 40 2 = 1.0826 2 2 (0) / 2 Early-time motion 2 4 6 8 10 Period 8 2.5 30 32 34 36 38 40 Fig. 12.8 Period Doubling Cascade Period doubling continues in a sequence of ever-closer values of Such period-doubling cascades are seen in many nonlinear systems Their form is essentially the same in all systems – it is “universal” Sub-harmonic frequency spectrum Driven Diode experiment F0 cos(t) /2 Period Doubling Cascade = 1.06 = 1.078 = 1.081 = 1.0826 ‘Bifurcation Points’ in the Period Doubling Cascade (0) / 2 Driven Damped Pendulum (0) 0 n period n 1 1→2 1.0663 2 3 4 2→4 4→8 8 → 16 1.0793 1.0821 1.0827 interval (n+1-n) 0.0130 0.0028 0.0006 The spacing between consecutive bifurcation points grows smaller at a steady rate: ( n 1 n ) 1 ( n n 1 ) ‘≈’ → ‘=’ as n → ∞ = 4.6692016 is called the Feigenbaum number The limiting value as n → ∞ is c = 1.0829. Beyond that is … chaos! Period infinity Chaos! = 1.105 (t) 5 10 15 20 25 30 t The pendulum is “trying” to oscillate at the driving frequency, but the motion remains erratic for all time Fig. 12.10 Period Doubling Cascade Period doubling continues in a sequence of ever-closer values of Such period-doubling cascades are seen in many nonlinear systems Their form is essentially the same in all systems – it is “universal” The Brain-behaviour Continuum: The Subtle Transition Between Sanity and Insanity By Jose Luis. Perez Velazquez Fig. 12.9, Taylor Chaos • Nonperiodic • Sensitivity to initial conditions Sensitivity of the Motion to Initial Conditions Start the motion of two identical pendulums with slightly different initial conditions Does their motion converge to the same attractor? Does it instead diverge quickly? Two pendulums 1 (t ), 2 (t ) are given different initial conditions Follow their evolution and calculate (t ) 2 (t ) 1 (t ) For a linear oscillator (t ) A cos(t ) C1er1t C2er2t Long-term attractor Transient behavior r1, 2 i1 The initial conditions affect the transient behavior, the long-term attractor is the same Hence (t ) De t cos(1t 1 ) Thus the trajectories will converge after the transients die out Convergence of Trajectories in Linear Motion (t ) De t cos(1t 1 ) Take the logarithm to magnify small differences.. ln[| (t ) |] ln(D) t | cos(1t 1 ) | Plotting log10[|(t)|] vs. t should be a straight line of slope –, plus some wiggles from the cos(1t – 1) term Note that log10[x] = log10[e] ln[x] Convergence of Trajectories in Linear Motion Log10[|(t)|] 2 2 4 4 6 8 10 t = 0.1 (0) = 0.1 Radians 6 8 10 12 The trajectories converge quickly for the small driving force (~ linear) case This shows that the linear oscillator is essentially insensitive to its initial conditions! Fig. 12.11 Convergence of Trajectories in Period-2 Motion Log10[|(t)|] 5 2 10 15 20 25 30 35 40 t = 1.07 (0) = 0.1 Radians 4 6 8 The trajectories converge more slowly, but still converge Fig. 12.12 Divergence of Trajectories in Chaotic Motion = 1.105 (0) = 0.0001 Radians Log10[|(t)|] 1 2 4 6 8 10 12 14 16 t 1 2 3 4 5 If the motion remains bounded, as it does in this case, then can never exceed 2. Hence this plot will saturate 6 The trajectories diverge, even when very close initially (16) ~ , so there is essentially complete loss of correlation between the pendulums Extreme Sensitivity to Initial Conditions Fig. 12.13 The Lyapunov Exponent (t ) ~ Ke t K 0 = Lyapunov exponent < 0: periodic motion in the long term > 0: chaotic motion Chaos • Nonperiodic • Sensitivity to initial conditions Linear Nonlinear Chaos Drive period Period doubling <0 Nonperiodic <0 >0 What Happens if we Increase the Driving Force Further? Does the chaos become more intense? (0) = 0.001 Radians (t) = 1.13 Log10[|(t)|] 5 10 15 20 t 2 3 t 5 10 6 15 9 Period 3 motion re-appears! Fig. 12.14 What Happens if we Increase the Driving Force Further? Does the chaos re-appear? (0) = 0.001 Radians = 1.503 (t) 5 10 20 10 15 Log10[|(t)|] t 20 25 1 1 2 3 4 5 10 15 t Chaotic motion re-appears! This is a kind of ‘rolling’ chaotic motion Fig. 12.15 Divergence of Two Nearby Initial Conditions for Rolling Chaotic Motion = 1.503 (t) 5 10 15 t 20 25 10 (0) = 0.001 Radians 20 Chaotic motion is always associated with extreme sensitivity to initial conditions Periodic and chaotic motion occur in narrow intervals of Fig. 12.16 Bifurcation Diagram Used to visualize the behavior as a function of driving amplitude 1) Choose a value of 2) Solve for (t), and plot a periodic sampling of the function (t0 ), (t0 1), (t0 2), (t0 3), (t0 4),... t0 chosen so that the attractor behavior is achieved 3) Move on to the next value of (0) / 2 (0) 0 1.0663 1.0793 Fig. 12.17 Construction of the Bifurcation Diagram (t) = 1.06 2 2.5 = 1.078 36 38 40 Period 2 32 34 36 38 40 t 2 2.5 = 1.081 Period 1 30 32 34 30 2 2.5 30 32 Period 4 34 36 38 40 Period 6 window = 1.0826 2 Period 8 2.5 30 32 34 36 38 40 The Rolling Motion Renders the Bifurcation Diagram Useless = 1.503 (t) 5 10 15 t 20 25 10 (0) = 0.001 Radians 20 As an alternative, plot (t ) Rolling Motion (next slide) Mostly chaos Period-1 followed by period doubling bifurcation Mostly chaos Period-3 Mostly chaos Previous diagram range (t ) Bifurcation Diagram Over a Broad Range of Period-1 Rolling Motion at = 1.4 = 1.4 (t ) (t) 5 10 t 10 5 10 10 20 20 Even though the pendulum is “rolling”, (t ) is periodic 10 t An Alternative View: State Space Trajectory Plot (t ) vs. (t ) with time as a parameter (t) 4 (0) / 2 (0) 0 1 2 (t ) t (t ) (t ) 4 First 20 cycles 6 4 start 4 5 periodic attractor 4 4 4 = 0.6 2 2 3 (t ) 4 4 4 Cycles 5 -20 Fig. 12.20, 12.21 An Alternative View: State Space Trajectory Plot (t ) vs. (t ) with time as a parameter = 0.6 (t ) (0) 0 (t ) (0) 0 4 4 (t ) start 4 4 4 (t ) 4 4 4 periodic attractor First 20 cycles The periodic attractor: [ (t ) ,(t ) ] is an ellipse Cycles 5 -20 (t ) A cos(t ) (t ) A sin(t ) The state space point moves clockwise on the orbit Fig. 12.22 State Space Trajectory for Period Doubling Cascade = 1.078 = 1.081 (t ) (t ) 10 10 (t ) 2 2 (t ) 2 10 2 10 Period-2 Period-4 Plotting cycles 20 to 60 Fig. 12.23 State Space Trajectory for Chaos (t ) = 1.105 Cycles 14 - 21 10 1 1 (t ) 10 (t ) 10 1 Cycles 14 - 94 The orbit has not repeated itself… 1 10 (t ) State Space Trajectory for Chaos = 1.5 = 0/8 Chaotic rolling motion Mapped into the interval – < < Cycles 10 – 200 This plot is still quite messy. There’s got to be a better way to visualize the motion … The Poincaré Section Similar to the bifurcation diagram, look at a sub-set of the data 1) Solve for (t), and construct the state-space orbit 2) Plot a periodic sampling of the orbit (t ),(t ), (t 0 0 0 1), (t0 1) , (t0 2), (t0 2) ,... with t0 chosen so that the attractor behavior is achieved = 1.5 = 0/8 Samples 10 – 60,000 Enlarged on the next slide The Poincaré Section is a Fractal The Poincaré section is a much more elegant way to represent chaotic motion The Superconducting Josephson Junction as a Driven Damped Pendulum 1 1 ei1 2 2 ei2 2 1 I (Tunnel barrier) The Josephson Equations 21 = phase difference of SC wave-function across the junction Chaos in Newtonian Billiards Imagine a point-particle trapped in a 2D enclosure and making elastic collisions with the walls s1 1 Describe the successive wall-collisions with a “mapping function” sn1 f (sn , n ) 0 n1 g (sn , n ) s0 Linear Maps for “Integrable” systems !! 0 sn1 f (sn , n ) n1 g (sn , n ) Non-Linear Maps for “Chaotic” systems !! • The “Chaos” arises due to the shape of the boundaries enclosing the system. Computer animation of extreme sensitivity to initial conditions for the stadium billiard