Java Applet Simulation and files related to the problem:
Approach to solving the problem
The second order differential equation governing the Spinning magnet is broken into two first order equations. One of them gives the rate of change of angle df/dt and the other is rate of change of corresponding canonical momenta dpf/dt . The evolution of these canonical variables gives us the phase plot. The given system is driven by an external magnetic field B0cos(wt). The system's periodic/quasi periodic/chaotic behavior strongly depends on the amplitude of magnetic field and applied frequency w. A plot of pf versus f at periodic intervals T=2p/w gives us the Poincare map. The critical parameter determining the chaotic behavior is l as defined above. l<1 gives periodic behavior, l>1 is chaotic and l=1 is quasi periodic motion. All the dynamics is controlled by two essential parameters B0m/I and w, so for computational purpose I treat as one constant A, and w as the other.
Phase and Poincare plot of Spinning magnet for various values of l are shown in figs. Phase plot shows angular velocity versus angle. Angle f is convoluted to modulo 2p . Poincare plot, plots the dpf/dt versus f at intervals of 2p/w.
Non chaotic region
Poincare map for l =0.5
It has three stable regions. As time progress it spends equal time in each of these basins. The period of this system is the number of distinct points plotted on all of these three loops. Such typical periodic/quasi behavior is shown for all l<1. For values l not equal to simple fraction or l near 1 the waiting time for periodic behavior is large. Observe below that l=1 shows two basins where the phase space is not accessed. These basins have very interesting mathematical properties.
At the transition l =1
Poincare map for l =1 Phase space plot for l =1
Transition to chaos l >1:
Poincare map for l =1.5 Phase space plot for l =1.5
Sensitivity to initial conditions
One of the most
important characteristic of any Non-linear chaotic system is its high
sensitivity to initial conditions. Small change in initial condition
will lead to entirely different trajectories in phase space. The above
spinning magnet system also shows such high degree of sensitivity to
Mathematically the difference b/w two functions diverge exponentially in the chaotic region.
ll is the Lyaponov exponent. If is ll< 1 Then it is non chaotic, ll>1 system is chaotic. I programmed to calculate Lyaponov exponent for a different system (Bifuracting system) and the result is plotted as below.
Lyaponov exponent for Bifurcation system as a function of r
Similar behavior is exhibited for spinning magnet.
The Poincare map clearly showed the transition to chaos when we varied the control parameter from l< 1 to l>1. Nonlinear effects are dominant in spinning magnet (in magnetic field) under certain conditions on its parameters.