Let a particle of mass *m* can move freely on the circle of length
*L _{x}*, and its coordinate

**Remark.**
The last assumption is not so artificial as one could think.
Say, in solid state physics, for quantum excitations (quasiparticles)
the notion of *quasimomentum* is used instead the ordinary momentum, and
the periodicity in respect to the quasimomentum takes place with a period
inversely proportional to spatial period of the crystal lattice.

If the particle has position and momentum *p* and *x*
at the moment before the *n-*th kick, then, just after the kick the position
remains yet the same, but the momentum becomes equal to
*p'*=*p*+g*x*.
Then, the particle moves by inertia
with velocity *p'*/*m*.
After time interval *T* the increase of the coordinate
will be
D*x*=*p'T*/*m*, and before the
next kick we have
*p'=p+gx* (mod *L _{p}*),

where the

or

The constructed mapping of variables *p* and
*x* is called **Arnold's cat map**.
This name is due to V.I.Arnold who explained the action of this map
using the picture of the cat face (see figure).
Geometrically, the first step of the procedure consists in linear transformation
of the variables

and the second one in transfer of elements of the picture
again inside of the unit square (the mod operation).
Due to periodicity in *x* and *p*,
the phase space may be regarded as the surface of a torus.
The particle motion is conservative, i.e. we deal with a Hamiltonian system.
Mathematically, it is connected with the fact that the determinant for the matrix
associated with the Arnold cat map equals 1, and it conserves the measure
(area) of any domain, say, of the cat face.
In terms of classic mechanics, this is the canonical
transformation.

One can consider more wide class of maps on the torus, which are determined by integer element matrices

,

satisfying the condition *ab-cd*=1.
In dependence on the eigenvalues
l_{1}, l_{2},
the maps of such kind relate to one of the three types:

The Arnold cat map is of the hyperbolic type
as the eigenvalues are
_{1}=(3+5^{1/2})/2_{2}=(3-5^{1/2})/2._{1}=logl_{1}
and
L_{2}=logl_{2}
are the *Lyapunov exponents*, one of which is positive.
This is the known sign of presence of sensitivity of motion to variation of
initial conditions, the important attribute of the dynamical chaos.

If we exclude the pulses of external field in the above consideration,
the result will be a parabolic type map describing free motion of the particle:
*p'=p*, *x'=p+x* (mod 1).

Canonical transformation swapping position and momentum
is an example of an elliptic map:
*p'=x*, *x'=-p* (mod 1),

In the figure we show evolution of some initial area under subsequent iterations of the maps of hyperbolic (), parabolic (b) and elliptic (c) type.

In the parabolic case the image of the figure remains in the definite initial interval of momentum. In the elliptic case the evolution consists in rotation without other changes of the form.

Under iterations of the hyperbolic map the cat face
is stretched along the unstable eigenvector, at each step by the factor
l_{1} and compressed along the
second (stable) eigenvector by the factor
l_{2}.
After a large number of iterations the cat face
transforms into an extremely narrow strip stretched along the
unstable direction, that is close to a long segment of straight line
*p=kx*, *k*=(5^{1/2}-1)/2.
As the coefficient is irrational, this line covers the
torus surface dense everywhere.
So, the picture looks like a set of a large number of the
narrow black and white strips, which arise from the cat face and its complimentary
set, respectively.
In the usual language, we should say that the "black" and "white"
liquids are well mixed.
The mixing property may be formulated mathematically. Presence of mixing
is proved rigorously for the hyperbolic maps and serves as
a basis for a conclusion on the chaotic nature of the dynamics.

Comparing figs (a)-(c), one can see that neither parabolic, nor elliptic maps manifest the mixing property.

of theoretical nonlinear

dynamics

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