Kneadings, Symbolic Dynamics and Painting Lorenz Chaos. A Tutorial

A new computational technique based on the symbolic description utilizing kneading invariants is proposed and verified for explorations of dynamical and parametric chaos in a few exemplary systems with the Lorenz attractor. The technique allows for uncovering the stunning complexity and universality of bi-parametric structures and detect their organizing centers - codimension-two T-points and separating saddles in the kneading-based scans of the iconic Lorenz equation from hydrodynamics, a normal model from mathematics, and a laser model from nonlinear optics.Comment: Journal of Bifurcations and Chaos, 201

Symbolic Toolkit for Chaos Explorations

New computational technique based on the symbolic description utilizing kneading invariants is used for explorations of parametric chaos in a two exemplary systems with the Lorenz attractor: a normal model from mathematics, and a laser model from nonlinear optics. The technique allows for uncovering the stunning complexity and universality of the patterns discovered in the bi-parametric scans of the given models and detects their organizing centers -- codimension-two T-points and separating saddles.Comment: International Conference on Theory and Application in Nonlinear Dynamics (ICAND 2012

Measuring the universal synchronization properties of coupled oscillators across the Hopf instability

When a driven oscillator loses phase-locking to a master oscillator via a Hopf bifurcation, it enters a bounded-phase regime in which its average frequency is still equal to the master frequency, but its phase displays temporal oscillations. Here we characterize these two synchronization regimes in a laser experiment, by measuring the spectrum of the phase fluctuations across the bifurcation. We find experimentally, and confirm numerically, that the low frequency phase noise of the driven oscillator is strongly suppressed in both regimes in the same way. Thus the long-term phase stability of the master oscillator is transferred to the driven one, even in the absence of phase-locking. The numerical study of a generic, minimal model suggests that such behavior is universal for any periodically driven oscillator near a Hopf bifurcation point.Comment: 5 pages, 5 figure

Transition from rotating waves to modulated rotating waves on the sphere

We study non-resonant and resonant Hopf bifurcation of a rotating wave in SO(3)-equivariant reaction-diffusion systems on a sphere. We obtained reduced differential equations on so(3), the characterization of modulated rotating waves obtained by Hopf bifurcation of a rotating wave, as well as results regarding the resonant case. Our main tools are the equivariant center manifold reduction and the theory of Lie groups and Lie algebras, especially for the group SO(3) of all rigid rotations on a sphere

Bifurcation analysis of a semiconductor laser with filtered optical feedback

We study the dynamics and bifurcations of a semiconductor laser with delayed filtered optical feedback, where a part of the output of the laser reenters after spectral filtering. This type of coherent optical feedback is more challenging than the case of conventional optical feedback from a simple mirror, but it provides additional control over the output of the semiconductor laser by means of choosing the filter detuning and the filter width. This laser system can be modeled by a system of delay differential equations with a single fixed delay, which is due to the travel time of the light outside the laser. In this paper we present a bifurcation analysis of the filtered feedback laser. We first consider the basic continuous wave states, known as the external filtered modes (EFMs), and determine their stability regions in the parameter plane of feedback strength versus feedback phase. The EFMs are born in saddle-node bifurcations and become unstable in Hopf bifurcations. We show that for small filter detuning there is a single region of stable EFMs, which splits up into two separate regions when the filter is detuned. We then concentrate on the periodic orbits that emanate from Hopf bifurcations. Depending on the feedback strength and the feedback phase, two types of oscillations can be found. First, there are undamped relaxation oscillations, which are typical for semiconductor laser systems. Second, there are oscillations with a period related to the delay time, which have the remarkable property that the laser frequency oscillates while the laser intensity is almost constant. These frequency oscillations are only possible due to the interaction of the laser with the filter. We determine the stability regions in the parameter plane of feedback strength versus feedback phase of the different types of oscillations. In particular, we find that stable frequency oscillations are dominant for nonzero values of the filter detuning. © 2007 Society for Industrial and Applied Mathematics

Stability of the self-phase-locked pump-enhanced singly resonant parametric oscillator

Steady-state and dynamics of the self-phase-locked (3\omega ==> 2\omega, \omega) subharmonic optical parametric oscillator are analyzed in the pump-and-signal resonant configuration, using an approximate analytical model and a full propagation model. The upper branch solutions are found always stable, regardless of the degree of pump enhancement. The domain of existence of stationary states is found to critically depend on the phase-mismatch of the competing second-harmonic process.Comment: LateX2e/RevteX4, 4 pages, 5 figures. Submitted to Phys. Rev. A (accepted on Jan. 17, 2003