Poincare Map and its application to 'Spinning Magnet' problem

Things should be made as simple as possible, but not anymore simpler.
-A. Einstein

A linear system when driven by an external periodic force will try to follow the applied force and hence oscillate with the applied frequency (after a short transient behavior). The situation is quite  different for a Nonlinear system. Here depending on the strength of nonlinearity, the system shows periodic or quasi periodic or chaotic behavior. We expect that in the periodic regime the period of system should be related to the period of external force. If T is the period of external force then system period will be nT, where n is an integer. In the chaotic regime n tends to infinity. The number distinct points(over a long period of time) plotted on Poincare map indicates the period of the system. Hence in chaotic regime the Poincare map tries to fill a subset of  phase space. Sometime it may so happen that Poincare map leaves out certain regions though it is accessible energetically. In such cases the system shows  a non-ergodic behavior.

A typical Poinacre map is shown in the following figure

Magnet in an oscillating magnetic field

A magnet in an oscillating magnetic field constitute one such system which can illustate interesting nonlinear behavior under Poincare
map.  The statement of the problem is as follows:

Consider a compass needle that is free to rotate in a periodically reversing magnetic field which is  perpendicular to the axis of the needle. The equation of motion of the needle is given by

Where   f is the angle of the needle with respect to a fixed axis along the field, m is the magnetic moment of the needle, I is its moment of inertia, and B0 and w are the amplitude and the angular frequency of the magnetic field, respectively. One can choose an appropriate numerical method for solving above equation and the Poincare map at time t = 2p/w.  One can verify that if the parameter
is greater than 1, then the system exhibits chaotic motion

Java Applet Simulation and files related to the problem:

The problem was simulated in java and the applet could be found at the following link:
PoincareMapApplet
Source code
A report regarding this problem, its implementation and results can be found at Spinning magnet

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.

Results

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 initial conditions.
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.

Conclusion:

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.