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Continuation of Bifurcations of Periodic Orbits for Large-Scale Systems

SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 2015年第2期14卷 674-698页

作者： Net, M. Sanchez, J. Univ Politecn Cataluna Dept Fis Aplicada ES-08034 Barcelona Spain

A methodology to track bifurcations of periodic orbits in large-scale dissipative systems depending on two parameters is presented. It is based on the application of iterative Newton-Krylov techniques to extended systems. To evaluate the action of the Jacobian it is necessary to integrate variational equations up to second order. It is shown that this is possible by integrating systems of dimension at most four times that of the original equations. In order to check the robustness of the method, the thermal convection of a mixture of two fluids in a rectangular domain has been used as a test problem. Several curves of codimension-one bifurcations, and the boundaries of an Arnold's tongue of rotation number 1/8, have been computed.

关键词： continuation methods numerical computation of invariant objects periodic orbits bifurcation tracking extended systems Newton-Krylov methods variational equations

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computation of Limit Cycles and Their Isochrons: Fast Algorithms and Their Convergence

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SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 2013年第4期12卷 1763-1802页

作者： Huguet, Gemma de la Llave, Rafael NYU Courant Inst Math Sci New York NY 10012 USA Georgia Inst Technol Sch Math Atlanta GA 30332 USA

We present efficient algorithms to compute limit cycles and their isochrons (i.e., the sets of points with the same asymptotic phase) for planar vector fields. We formulate a functional equation for the parameterization of the invariant cycle and its isochrons, and we show that it can be solved by means of a Newton method. Using the right transformations, we can solve the equation of the Newton step efficiently. The algorithms are efficient in the sense that if we discretize the functions using N points, a Newton step requires O(N) storage and O(N log N) operations in Fourier discretization or O(N) operations in other discretizations. We prove convergence of the algorithms and present a validation theorem in an a posteriori format. That is, we show that if there is an approximate solution of the invariance equation that satisfies some some mild nondegeneracy conditions, then there is a true solution nearby. Thus, our main theorem can be used to validate numerically computed solutions. The theorem also shows that the isochrons are analytic and depend analytically on the base point. Moreover, it establishes smooth dependence of the solutions on parameters and provides efficient algorithms to compute perturbative expansions with respect to external parameters. We include a discussion on the numerical implementation of the algorithms as well as numerical results for representative examples.

关键词： parameterization method isochrons invariant manifolds numerical computation of invariant objects phase resetting curves biological oscillators

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A computational and Geometric Approach to Phase Resetting Curves and Surfaces

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SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 2009年第3期8卷 1005-1042页

作者： Guillamon, Antoni Huguet, Gemma Univ Politecn Cataluna Dept Matemat Aplicada 1 E-08028 Barcelona Catalonia Spain Ctr Recerca Matemat E-08193 Barcelona Catalonia Spain

This work arises from the purpose of applying new tools in dynamical systems to time problems in biological systems. The main aim of this paper is to develop a numerical method to perform the effective computation of the phase advancement when we stimulate an oscillator which has not yet reached the asymptotic state (a limit cycle). That is, we want to extend the computation of the phase resetting curves (PRCs) (the classical tool to compute the phase advancement) to a neighborhood of the limit cycle, obtaining what we call the phase resetting surfaces (PRSs). To achieve this goal we first perform a careful study of the theoretical grounds (the parameterization method for invariant manifolds and another approach using Lie symmetries), which allows us to describe the isochronous sections of the limit cycle and, from them, to obtain the PRSs. In order to make this theoretical framework applicable, we use the numerical algorithms of the parameterization method and other semianalytical tools to extend invariant manifolds;as a result, we design a numerical scheme to compute both the isochrons and the PRSs of a given oscillator. Finally, to illustrate this algorithm, we apply it to some well-known biological models and we include a discussion on different biological and numerical aspects suggested by these examples.

关键词： parameterization method Lie symmetries isochrons phase resetting curves numerical computation of invariant objects biological oscillators

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