Universality and Fractal Dimension of Mode Locking Structure in Systems with Competing Periodicities

摘要

We are concerned with the properties of physical systems with competing periodicities. Two very different situations are considered. In the first case the competition is between two periods in time, or two frequencies, in a system which is at the verge of entering into a chaotic state. In the second case the competition is between two periods in space. In both cases there is a nonlinear coupling which tend to lock the two periods together. An example of a system of the first type is a driven damped pendulum perturbed by an external oscillating force. In the absence of the periodic force the pendulum will oscillate, or rotate with a frequency of its own. If the coupling is weak the frequencies will lock only if the unperturbed frequencies are related to each other through almost rational numbers. If the coupling is stronger there will be an increased tendence towards locking. At some critical strength the modes may always be locked so that the resulting frequency is always a rational fraction of the frequency of the external perturbation. If the coupling is even stronger the motion may become chaotic, with no single basic frequency. In the second case one has a competition between the period, or wave vector of the spatially ordered structure, such as a spin-density wave, and the periodicity of the lattice on which the system is defined. A specific model of this type is the axial next-nearest-neighbor Ising (ANNNI) model. For some values of the parameters the magnetic structure has a period of its own, not related to the periodicity of the underlying lattice. In other regimes of the phase diagram, where the effective coupling is stronger, the resulting structure is always commensurable with the lattice. As a specific example we consider a simple one-dimensional Ising model with long range antiferromagnetic interactions. (ERA citation 08:055217)