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Journal of Computational Dynamics

June & December 2018 , Volume 5 , Issue 1&2

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Optimal transport over nonlinear systems via infinitesimal generators on graphs
Karthik Elamvazhuthi and Piyush Grover
2018, 5(1&2): 1-32 doi: 10.3934/jcd.2018001 +[Abstract](2182) +[HTML](237) +[PDF](8429.96KB)
Abstract:

We present a set-oriented graph-based computational framework for continuous-time optimal transport over nonlinear dynamical systems. We recover provably optimal control laws for steering a given initial distribution in phase space to a final distribution in prescribed finite time for the case of non-autonomous nonlinear control-affine systems, while minimizing a quadratic control cost. The resulting control law can be used to obtain approximate feedback laws for individual agents in a swarm control application. Using infinitesimal generators, the optimal control problem is reduced to a modified Monge-Kantorovich optimal transport problem, resulting in a convex Benamou-Brenier type fluid dynamics formulation on a graph. The well-posedness of this problem is shown to be a consequence of the graph being strongly-connected, which in turn is shown to result from controllability of the underlying dynamical system. Using our computational framework, we study optimal transport of distributions where the underlying dynamical systems are chaotic, and non-holonomic. The solutions to the optimal transport problem elucidate the role played by invariant manifolds, lobe-dynamics and almost-invariant sets in efficient transport of distributions in finite time. Our work connects set-oriented operator-theoretic methods in dynamical systems with optimal mass transportation theory, and opens up new directions in design of efficient feedback control strategies for nonlinear multi-agent and swarm systems operating in nonlinear ambient flow fields.

Balanced model order reduction for linear random dynamical systems driven by Lévy noise
Martin Redmann and Melina A. Freitag
2018, 5(1&2): 33-59 doi: 10.3934/jcd.2018002 +[Abstract](245) +[HTML](148) +[PDF](947.38KB)
Abstract:

When solving linear stochastic differential equations numerically, usually a high order spatial discretisation is used. Balanced truncation (BT) and singular perturbation approximation (SPA) are well-known projection techniques in the deterministic framework which reduce the order of a control system and hence reduce computational complexity. This work considers both methods when the control is replaced by a noise term. We provide theoretical tools such as stochastic concepts for reachability and observability, which are necessary for balancing related model order reduction of linear stochastic differential equations with additive Lévy noise. Moreover, we derive error bounds for both BT and SPA and provide numerical results for a specific example which support the theory.

Computer-assisted proofs for radially symmetric solutions of PDEs
István Balázs, Jan Bouwe van den Berg, Julien Courtois, János Dudás, Jean-Philippe Lessard, Anett Vörös-Kiss, JF Williams and Xi Yuan Yin
2018, 5(1&2): 61-80 doi: 10.3934/jcd.2018003 +[Abstract](275) +[HTML](181) +[PDF](1052.44KB)
Abstract:

We obtain radially symmetric solutions of some nonlinear (geometric) partial differential equations via a rigorous computer-assisted method. We introduce all main ideas through examples, accessible to non-experts. The proofs are obtained by solving for the coefficients of the Taylor series of the solutions in a Banach space of geometrically decaying sequences. The tool that allows us to advance from numerical simulations to mathematical proofs is the Banach contraction theorem.

The snapback repellers for chaos in multi-dimensional maps
Kang-Ling Liao, Chih-Wen Shih and Chi-Jer Yu
2018, 5(1&2): 81-92 doi: 10.3934/jcd.2018004 +[Abstract](67) +[HTML](31) +[PDF](392.33KB)
Abstract:

The key of Marotto's theorem on chaos for multi-dimensional maps is the existence of snapback repeller. For practical application of the theory, locating a computable repelling neighborhood of the repelling fixed point has thus become the key issue. For some multi-dimensional maps \begin{document}$F$\end{document}, basic information of \begin{document}$F$\end{document} is not sufficient to indicate the existence of snapback repeller for \begin{document}$F$\end{document}. In this investigation, for a repeller \begin{document}$\bar{\bf z}$\end{document} of \begin{document}$F$\end{document}, we start from estimating the repelling neighborhood of \begin{document}$\bar{\bf z}$\end{document} under \begin{document}$F^{k}$\end{document} for some \begin{document}$k ≥ 2$\end{document}, by a theory built on the first or second derivative of \begin{document}$F^k$\end{document}. By employing the Interval Arithmetic computation, we locate a snapback point \begin{document}${\bf z}_0$\end{document} in this repelling neighborhood and examine the nonzero determinant condition for the Jacobian of \begin{document}$F$\end{document} along the orbit through \begin{document}${\bf z}_0$\end{document}. With this new approach, we are able to conclude the existence of snapback repellers under the valid definition, hence chaotic behaviors, in a discrete-time predator-prey model, a population model, and the FitzHugh nerve model.

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