Harper Map

After downloading harper.jar, please execute it by double-clicking, or typing "java -jar harper.jar".

By clicking your mouse, you can set the initial value of a time evolution.
By dragging your mouse, you can expand the field.

If the above application does not start, please install OpenJDK from adoptium.net.

In this page, we treat the Harper map written as

xn+1 = xn + (K/2π) sin (2π yn) modulo 1
yn+1 = yn - (L/2π) sin (2π xn+1) modulo 1.

Like the Standard map, the determinant of the Jacobian matrix of this map takes one, thus the area of the phase space is preserved. The parameters K and L denote the strength of the nonlinearity.

This model is obtained from the Hamiltonian of a Bloch electron in the presence of a magnetic field and subject to an alternating electromagnetic field.
Applying the same Hamiltonian to the Schrödinger equation, the relation with quantum chaos is also discussed.

By increasing the nonlinear parameters K and L, the surviving tori are destructed, and the chaotic time evolutions are observed.
For example, with K=L=5.0, islands of tori are invisible.

However, with particular values of K and L (e.g., K=L=6.34), the new tori are created by a bifurcation.
This tori are called as accelerator tori and cause the anomalous diffusion of the observables.

The accelerator tori and the anomalous diffusion are also observed in Standard map.
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Introduction to Chaos and Nonlinear Dynamics