Periodically driven oscillator with nonmonotonic
dependence of eigenfrequency on energy
S.M. Soskin
Institute of Semiconductor Physics, National Ukrainian Academy
of Sciences, Kiev, Ukraine
Studies of periodically driven systems have very long history. In particular,
the concept of nonlinear resonance in driven Hamiltonian systems became widespread
after the famous paper by Chirikov [Chirikov, 1979]. In the simplest case, of a
one-dimensional nonlinear oscillator subject to a weak periodic force, it means
that, besides the linear (i.e. weak) response of the oscillator, the strongly
nonlinear one could exist which is equivalent approximately to an eigenoscilation
at the resonant energy i.e. at such energy 
for which a frequency of an
eigenoscillation 
is equal to the frequency of the driving force.
At more careful consideration, it turns out that the response amplitude oscillates
slowly near the resonant value. These oscillations are similar to pendulum
oscillations. The condition for just such kind of oscillations to be realised is
the so called condition of a moderate nonlinearity [Chirikov, 1979]. The case
of a weak nonlinearity was studied for multi-dimensional intrinsically degenerated
systems [Jaeger and Lichtenberg, 1970] and for nearly linear one-dimensional oscillator
[Israelev, 1980].
Strange, however, that an oscillator for which 
possesses
an extremum while the driving frequency is close to the extremal eigenfrequency,
has not been studied until very recently. In the first paper devoted to this case
[Soskin, 1994], the main emphasis was made at the counterintuitive feature of such a
system, namely a possibility of the strongly nonlinear response without an exact matching
between the driving frequency and a frequency of any eigenoscillation. After this
paper, such systems were intensively studied. It turned out that the nonlinear resonance
in them differed in many respects from that one in the conventional case (NR),
so that it got a special name, zero-dispersion nonlinear resonance (ZDNR).
In the present work, the current status of research on ZDNR an related phenomena is reviewed. The main emphasises are made at 1) dramatic jump-wise changes in the fluctuational inter-attractor transition probabilities as parameters of the driving force change which are associated with ZDNR/NR transition and some certain other global bifurcations in the system [Luchinsky et al, 1995], 2) very interesting manifestations of chaos in the system in the presence of dissipation [ Soskin et al, 1995].
In conclusion, I would like to note that ZDNR and related phenomena can occur in many physical systems, e.g. SQUIDs, relativistic nonlinear oscillators, electric circuits.