Complex Dynamics and Synchronization of Delayed-Feedback Nonlinear Oscillators

Abstract

We describe a flexible and modular delayed-feedback nonlinear oscillator that is capable of generating a wide range of dynamical behaviours, from periodic oscillations to high-dimensional chaos. The oscillator uses electrooptic modulation and fibre-optic transmission, with feedback and filtering implemented through real-time digital-signal processing. We consider two such oscillators that are coupled to one another, and we identify the conditions under which they will synchronize. By examining the rates of divergence or convergence between two coupled oscillators, we quantify the maximum Lyapunov exponents or transverse Lyapunov exponents of the system, and we present an experimental method to determine these rates that does not require a mathematical model of the system. Finally, we demonstrate a new adaptive control method that keeps two oscillators synchronized even when the coupling between them is changing unpredictably.

24 pages, 13 figures. To appear in Phil. Trans. R. Soc. A (special theme issue to accompany 2009 International Workshop on Delayed Complex Systems)

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• Delayed Dynamical Systems: Networks, Chimeras and Reservoir Computing
• Harvesting entropy and quantifying the transition from noise to chaos in a photon-counting feedback loop
• Resumption of dynamism in damaged networks of coupled oscillators
• Physical deep learning based on optimal control of dynamical systems
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• Estimation of communication-delays through adaptive synchronization of chaos
• Machine Learning Link Inference of Noisy Delay-coupled Networks with Opto-Electronic Experimental Tests
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• Highly-complex optical signal generation using electro-optical systems with non-linear, non-invertible transmission functions
• Dynamic synchronization of a time-evolving optical network of chaotic oscillators
• Dynamic environment coupling induce synchronized states in coupled time-delayed electronic circuits
• Chaos synchronization by resonance of multiple delay times
• Blending Optimal Control and Biologically Plausible Learning for Noise-Robust Physical Neural Networks
• Synchronization of Chaos