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Renato C. Calleja Eusebius J. Doedel Carlos García-Azpeitia 《Regular and Chaotic Dynamics》2018,23(5):595-612
We consider the equations of motion of n vortices of equal circulation in the plane, in a disk and on a sphere. The vortices form a polygonal equilibrium in a rotating frame of reference. We use numerical continuation in a boundary value setting to determine the Lyapunov families of periodic orbits that arise from the polygonal relative equilibrium. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship, the orbit is also periodic in the inertial frame. A dense set of Lyapunov orbits, with frequencies satisfying a Diophantine equation, corresponds to choreographies of n vortices. We include numerical results for all cases, for various values of n, and we provide key details on the computational approach. 相似文献
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A successive continuation method for locating connecting orbits in parametrized systems of autonomous ODEs is considered.
A local convergence analysis is presented and several illustrative numerical examples are given.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
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We investigate the bifurcation structure of a family of relative equilibria of a ring of seven oscillators described by the discrete nonlinear Schrödinger equation (DNLSE) when the period of these orbits and a suitable defect act as bifurcation parameters. We find a reduced Hamiltonian that gives substantial insight into the dynamics of this system. The convexity of this Hamiltonian at given nonresonant equilibria supports the stability of nearby quasiperiodic solutions. We show that the local loss of convexity in the reduced Hamiltonian is determined by the Hessian of its integrable part in the family of relative equilibria under study. Stable quasiperiodic solutions are studied by considering the power spectral densities of a set of suitable fast and slow actions, whose origin is suggested by the averaging principle. We also show that the return times form an optimal embedding to characterize the system dynamics. We show that the power spectral density of a suitable interference signal, arising from a ring of Bose-Einstein condensates and described by the DNLSE, has a single prominent peak at the breather-like relative equilibria. 相似文献
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Summary. We prove numerical stability of a class of piecewise polynomial collocation methods on nonuniform meshes for computing asymptotically
stable and unstable periodic solutions of the linear delay differential equation by a (periodic) boundary value approach. This equation arises, e.g., in the study of the numerical stability of collocation
methods for computing periodic solutions of nonlinear delay equations. We obtain convergence results for the standard collocation
algorithm and for two variants. In particular, estimates of the difference between the collocation solution and the true solution
are derived. For the standard collocation scheme the convergence results are “unconditional”, that is, they do not require
mesh-ratio restrictions. Numerical results that support the theoretical findings are also given.
Received June 9, 2000 / Revised version received December 14, 2000 / Published online October 17, 2001 相似文献
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Eusebius J. Doedel Carlos L. Pando Lambruschini 《The European physical journal. Special topics》2016,225(13-14):2613-2622
We study a rate-equation model for two coupled molecular lasers with a saturable absorber. A numerical bifurcation study shows the existence of isolas for in-phase periodic solutions as physical parameters change. In addition there are other non-isola families of in-phase, anti-phase and intermediate-phase periodic oscillations. In this model the unstable periodic orbits belonging to the in-phase isolas constitute a skeleton of the attractor, when chaotic synchronization sets in for a set of physically relevant control parameters. 相似文献