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In two-dimensional reaction-diffusion systems, local curvature perturbations on traveling waves are typically damped out and vanish. However, if the inhibitor diffuses much faster than the activator, transversal instabilities can arise, leading from flat to folded, spatio-temporally modulated waves and to spreading spiral turbulence. Here, we propose a scheme to induce or inhibit these instabilities via a spatio-temporal feedback loop. In a piecewise-linear version of the FitzHugh-Nagumo model, transversal instabilities and spiral turbulence in the uncontrolled system are shown to be suppressed in the presence of control, thereby stabilizing plane wave propagation. Conversely, in numerical simulations with the modified Oregonator model for the photosensitive Belousov-Zhabotinsky reaction, which does not exhibit transversal instabilities on its own, we demonstrate the feasibility of inducing transversal instabilities and study the emerging wave patterns in a well-controlled manner.
We demonstrate that a multiple delayed feedback is a powerful tool to control coherence properties of autonomous self-sustained oscillators. We derive the equation for the phase dynamics in presence of noise and delay, and analyze it analytically. In Gaussian approximation a closed set of equations for the frequency and the diffusion constant is obtained. Solutions of these equations are in good agreement with direct numerical simulations.
In the present work, we use symbolic regression for automated modeling of dynamical systems. Symbolic regression is a powerful and general method suitable for data-driven identification of mathematical expressions. In particular, the structure and parameters of those expressions are identified simultaneously.
We consider two main variants of symbolic regression: sparse regression-based and genetic programming-based symbolic regression. Both are applied to identification, prediction and control of dynamical systems.
We introduce a new methodology for the data-driven identification of nonlinear dynamics for systems undergoing abrupt changes. Building on a sparse regression algorithm derived earlier, the model after the change is defined as a minimum update with respect to a reference model of the system identified prior to the change. The technique is successfully exemplified on the chaotic Lorenz system and the van der Pol oscillator. Issues such as computational complexity, robustness against noise and requirements with respect to data volume are investigated.
We show how symbolic regression can be used for time series prediction. Again, issues such as robustness against noise and convergence rate are investigated us- ing the harmonic oscillator as a toy problem. In combination with embedding, we demonstrate the prediction of a propagating front in coupled FitzHugh-Nagumo oscillators. Additionally, we show how we can enhance numerical weather predictions to commercially forecast power production of green energy power plants.
We employ symbolic regression for synchronization control in coupled van der Pol oscillators. Different coupling topologies are investigated. We address issues such as plausibility and stability of the control laws found. The toolkit has been made open source and is used in turbulence control applications.
Genetic programming based symbolic regression is very versatile and can be adapted to many optimization problems. The heuristic-based algorithm allows for cost efficient optimization of complex tasks.
We emphasize the ability of symbolic regression to yield white-box models. In contrast to black-box models, such models are accessible and interpretable which allows the usage of established tool chains.
Standing waves are studied as solutions of a complex Ginsburg-Landau equation subjected to local and global time-delay feedback terms. The onset of standing waves is studied at the instability of the homogeneous periodic solution with respect to spatially periodic perturbations. The solution of this spatiotemporal wave pattern is given and is compared to the homogeneous periodic solution.