Next: References Up: Hamiltonian Dynamics of Previous: The stability analysis
Next: References
Up: Hamiltonian
Dynamics of Previous: The
stability analysis
We have shown that ferroelectric liquid crystals is a non-integrable Hamiltonian system, capable to exhibit chaos and which can be studied in laboratory.
Other real physical systems with these characteristics are the hydrogen atom in an uniform magnetic field, [16], and ballistic electrons in semiconductor micro structures (lateral surface super-lattices), [17], whose chaotic behavior has been intensively studied.
The experimental study of the chaotic molecular configuration
of (DOBAMBC) (for example), together with the mathematical methods developed
to study chaotic systems, [18, 19],
may help us to understand the irregularities experimentally observed in
the boundary lines of the 
diagram near the Lifshitz point.
Indeed, the functional dependence of the boundary - lines, on T and
, obtained from the stability analysis, is valid only bellow a critical
field 
, beyond which the chaotic solutions are dominant. The form of these lines
are related with the eigenvalues of the stability matrix. In the
chaotic region, these eigenvalues are very sensitive to the initial conditions
and their fluctuations generates the multifractal structure of the
Poincaré section [20]. Thus the computation
of the Liapunov exponents, [21], for
the present Hamiltonian will be very useful in the description of the phase
diagram in the neighbourhood of the Lifshitz point.
Another aspect that we want to call attention is for the universality in the transition to chaos.
It is well established experimentally that the universal cross over
exponent (that govern the 
- line, near the Lifshitz points) is 
, [22], according to renormalization theory
for critical phenomenon.
We observe that the quasi-periodic transition to chaos with the breakup of the KAM torus is just the reminiscence of the second order phase transition in critical phenomenon. Then we expect that the renormalization scheme for area - preserving maps, [23, 24], as for example the standard map
at 
, for the golden frequency 
, may furnish the correct behavior of the boundaries near the Lifshitz
point.
The global mode - locking of the map at 
for different values of 
, should put in evidence the Hamiltonian Devil's Staircase in ferroelectric
liquid crystals.