Journal of Computational Dynamics
http://aimsciences.org/
Kernel methods for the approximation of some key quantities of nonlinear systems
http://aimsciences.org//article/id/c82c8e96-a7aa-4a0f-a6d0-b05c6887bacc
We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control systems and random nonlinear dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systems -with a reasonable expectation of success -once the nonlinear system has been mapped into a high or infinite dimensional feature space. In particular, we embed a nonlinear system in a reproducing kernel Hilbert space where linear theory can be used to develop computable, non-parametric estimators approximating controllability and observability energy functions for nonlinear systems. In all cases the relevant quantities are estimated from simulated or observed data. It is then shown that the controllability energy estimator provides a key means for approximating the invariant measure of an ergodic, stochastically forced nonlinear system.]]>41119
Bouvrie Jake, Boumediene Hamzi
Parameterization method for unstable manifolds of delay differential equations
http://aimsciences.org//article/id/c52b3493-dc73-4956-9713-38d1589eec33
This work is concerned with efficient numerical methods for computing high order Taylor and Fourier-Taylor approximations of unstable manifolds attached to equilibrium and periodic solutions of delay differential equations. In our approach we first reformulate the delay differential equation as an ordinary differential equation on an appropriate Banach space. Then we extend the Parameterization Method for ordinary differential equations so that we can define operator equations whose solutions are charts or covering maps for the desired invariant manifolds of the delay system. Finally we develop formal series solutions of the operator equations. Order-by-order calculations lead to linear recurrence equations for the coefficients of the formal series solutions. These recurrence equations are solved numerically to any desired degree. The method lends itself to a-posteriori error analysis, and recovers the dynamics on the manifold in addition to the embedding. Moreover, the manifold is not required to be a graph, hence the method is able to follow folds in the embedding. In order to demonstrate the utility of our approach we numerically implement the method for some 1, 2, 3 and 4 dimensional unstable manifolds in problems with constant, and (briefly) state dependent delays.]]>41150
C. M. Groothedde, J. D. Mireles James
Rigorous continuation of bifurcation points in the diblock copolymer equation
http://aimsciences.org//article/id/77185b5a-2714-4516-b7e2-91d39f3dafbd
We develop general methods for rigorously computing continuous branches of bifurcation points of equilibria, specifically focusing on fold points and on pitchfork bifurcations which are forced through \begin{document}${\mathbb{Z}}_2$\end{document} symmetries in the equation. We apply these methods to secondary bifurcation points of the one-dimensional diblock copolymer model.]]>41148
Jean-Philippe Lessard, Evelyn Sander, Thomas Wanner
Set-oriented numerical computation of rotation sets
http://aimsciences.org//article/id/0a73bee3-940e-4984-9a2b-f88be31a26dd
We establish a set-oriented algorithm for the numerical approximation of the rotation set of homeomorphisms of the two-torus homotopic to the identity. A theoretical background is given by the concept of \begin{document}$\varepsilon$\end{document}-rotation sets. These are obtained by replacing orbits with \begin{document}$\varepsilon$\end{document}-pseudo-orbits in the definition of the Misiurewicz-Ziemian rotation set and are shown to converge to the latter as \begin{document}$\varepsilon$\end{document} decreases to zero. Based on this result, we prove the convergence of the numerical approximations as precision and iteration time tend to infinity. Further, we provide analytic error estimates for the algorithm under an additional boundedness assumption, which is known to hold in many relevant cases and in particular for non-empty interior rotation sets.]]>41123
Katja Polotzek, Kathrin Padberg-Gehle, Tobias Jäger