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Register a new userPhenomena of complex analytic dynamics in the non-autonomous, nonlinear ring system
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The model system manifesting phenomena peculiar to complex analytic maps is offered. The system is a non-autonomous ring cavity with nonlinear elements and filters,
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A feasible model is introduced that manifests phenomena intrinsic to iterative complex analytic maps (such as the Mandelbrot set and Julia sets). The system is composed of two coupled alternately excited oscillators (or self-sustained oscillators). The idea is based on a turn-by-turn transfer of the excitation from one subsystem to another (S.P.~Kuznetsov, Phys.~Rev.~Lett. bf 95 rm, 2005, 144101) accompanied with appropriate nonlinear transformation of the complex amplitude of the oscillations in the course of the process. Analytic and numerical studies are performed. Special attention is paid to an analysis of the violation of the applicability of the slow amplitude method with the decrease in the ratio of the period of the excitation transfer to the basic period of the oscillations. The main effect is the rotation of the Mandelbrot-like set in the complex parameter plane; one more effect is the destruction of subtle small-scale fractal structure of the set due to the presence of non-analytic terms in the complex amplitude equations.
The possibility of realization of the phenomena of complex analytic dynamics for the realistic physical models are investigated. Observation of the Mandelbrot and Julia sets in the parameter and phase spaces both for the discrete maps and non-autonomous continuous systems is carried out. For these purposes, the method, based on consideration of coupled systems, demonstrating period-doubling cascade is suggested. Novel mechanism of synchronization loss in coupled systems corresponded to the dynamical behavior intrinsic to the complex analytic maps is offered.
Many important high-dimensional dynamical systems exhibit complex chaotic behaviour. Their complexity means that their dynamics are necessarily comprehended under strong reducing assumptions. It is therefore important to have a clear picture of these reducing assumptions range of validity. The highly influential chaotic hypothesis of Gallavotti and Cohen states that the large-scale dynamics of high-dimensional systems are effectively hyperbolic, which implies many felicitous statistical properties. We demonstrate, contrary to the chaotic hypothesis, the existence of non-hyperbolic large-scale dynamics in a mean-field coupled system. To do this we reduce the system to its thermodynamic limit, which we approximate numerically with a Chebyshev Galerkin transfer operator discretisation. This enables us to obtain a high precision estimate of a homoclinic tangency, implying a failure of hyperbolicity. Robust non-hyperbolic behaviour is expected under perturbation. As a result, the chaotic hypothesis should not be assumed to hold in all systems, and a better understanding of the domain of its validity is required.
Silicon crystal puller (SCP) is a key equipment in silicon wafer manufacture, which is, in turn, the base material for the most currently used integrated circuit (IC) chips. With the development of the techniques, the demand for longer mono-silicon crystal rod with larger diameter is continuously increasing in order to reduce the manufacture time and the price of the wafer. This demand calls for larger SCP with increasing height, however, it causes serious swing phenomenon of the crystal seed. The strong swing of the seed causes difficulty in the solidification and increases the risk of mono-silicon growth failure.The main aim of this paper is to analyze the nonlinear dynamics in the FSRL system of the SCP. A mathematical model for the swing motion of the FSRL system is derived. The influence of relevant parameters, such as system damping, excitation amplitude and rotation speed, on the stability and the responses of the system are analyzed. The stability of the equilibrium, bifurcation and chaotic motion are demonstrated, which are often observed in practical situations. Melnikov method is used to derive the possible parameter region that leads to chaotic motion. Three routes to chaos are identified in the FSRL system, including period doubling, symmetry-breaking bifurcation and interior crisis. The work in this paper explains the complex dynamics in the FSRL system of the SCP, which will be helpful for the SCP designers in order to avoid the swing phenomenon in the SCP.
We revisit the stabilization of ionization of atoms subjected to a superintense laser pulse using nonlinear dynamics. We provide an explanation for the lack of complete ionization at high intensity and for the decrease of the ionization probability as intensity is increased. We investigate the role of each part of the laser pulse (ramp-up, plateau, ramp-down) in this process. We emphasize the role of the choice for the ionization criterion, energy versus distance criterion.
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