The Existence of Transverse Homoclinic Points in the Sitnikov Problem

AbstractUsing Melnikov′s method we are able to prove the existence of transverse homoclinic orbits and therefore the existence of a horseshoe in a special restricted three-body problem. This analysis is an alternative to the one described by Moser ("Stable and Random Motions in Dynamical Systems," Princeton Univ. Press, Princeton, NJ, 1973), based on Sitnikov′s original work (Dokl. Akad. Nauk. USSR 133, No. 2 (1960), 303-306), where the task is accomplished using a more direct construction of the horseshoe

Optimization along families of periodic and quasiperiodic orbits in dynamical systems with delay

This is the final version. Available on open access from Springer Verlag via the DOI in this recordThis paper generalizes a previously-conceived, continuation-based optimization technique for scalar objective functions on constraint manifolds to cases of periodic and quasiperiodic solutions of delay-differential equations. A Lagrange formalism is used to construct adjoint conditions that are linear and homogenous in the unknown Lagrange multipliers. As a consequence, it is shown how critical points on the constraint manifold can be found through several stages of continuation along a sequence of connected one-dimensional manifolds of solutions to increasing subsets of the necessary optimality conditions. Due to the presence of delayed and advanced arguments in the original and adjoint differential equations, care must be taken to determine the degree of smoothness of the Lagrange multipliers with respect to time. Such considerations naturally lead to a formulation in terms of multi-segment boundary-value problems (BVPs), including the possibility that the number of segments may change, or that their order may permute, during continuation. The methodology is illustrated using the software package coco on periodic orbits of both linear and nonlinear delay-differential equations, keeping in mind that closed-form solutions are not typically available even in the linear case. Finally, we demonstrate optimization on a family of quasiperiodic invariant tori in an example unfolding of a Hopf bifurcation with delay and parametric forcing. The quasiperiodic case is a further original contribution to the literature on optimization constrained by partial differential BVPs.Engineering and Physical Sciences Research Council (EPSRC)European Union Horizon 202

Sensitivity analysis for periodic orbits and quasiperiodic invariant tori using the adjoint method

This is the author accepted manuscript. The final version is available from the American Institute of Mathematical Sciences via the DOI in this recordCode availability: The code included in this paper constitutes fully executable scripts. Complete code, including that used to generate the results in Fig. 1, is available at https://github.com/jansieber/adjoint-sensitivity2022-supp.This paper presents a rigorous framework for the continuation of solutions to nonlinear constraints and the simultaneous analysis of the sensitivities of test functions to constraint violations at each solution point using an adjoint-based approach. By the linearity of a problem Lagrangian in the associated Lagrange multipliers, the formalism is shown to be directly amenable to analysis using the COCO software package, specifically its paradigm for staged problem construction. The general theory is illustrated in the context of algebraic equations and boundary-value problems, with emphasis on periodic orbits in smooth and hybrid dynamical systems, and quasiperiodic invariant tori of flows. In the latter case, normal hyperbolicity is used to prove the existence of continuous solutions to the adjoint conditions associated with the sensitivities of the orbital periods to parameter perturbations and constraint violations, even though the linearization of the governing boundary-value problem lacks a bounded inverse, as required by the general theory. An assumption of transversal stability then implies that these solutions predict the asymptotic phases of trajectories based at initial conditions perturbed away from the torus. Example COCO code is used to illustrate the minimal additional investment in setup costs required to append sensitivity analysis to regular parameter continuation.Engineering and Physical Sciences Research Council (EPSRC

An Extended Continuation Problem for Bifurcation Analysis

ABSTRACT This paper presents an extended formulation of the basic continuation problem for implicitly-defined, embedded manifolds in R n . The formulation is chosen so as to allow for the arbitrary imposition of additional constraints during continuation and the restriction to selective parametrizations of the corresponding higher-co-dimension solution manifolds. In particular, the formalism is demonstrated to clearly COPYRIGHT This is a preprint of a paper originally published by ASME: Dankowicz, H., Schilder, F., "An Extended Continuation Problem for Bifurcation Analysis in the Presence of Constraints," Journal on Computational and Nonlinear Dynamics, to appear. c 2010 ASME Publishing, Three Park Avenue, New York, NY 10016. Introduction Continuation is a numerical technique for computing implicitly-defined manifolds that relies on the Implicit Function Theorem (IFT) and its constructive proof. Starting with a single chart, i.e., a point on the manifold together with a representation of the tangent space at this point, continuation employs a covering algorithm for computing nearby charts. The process is subsequently repeated for each of the nearby charts. The manifold through the initial point is called a branch or a family and the computed atlas of charts is a covering of this branch. General-purpose covering algorithms were first developed for the case of one-dimensional manifolds, the most successful one being the pseudo-arc length continuation method It is instructive to distinguish between three different layers of an application-oriented implementation of a continuation problem, a distinction that has been made in all modern continuation packages. At the core layer one finds the covering algorithm and other general-purpose tools that provide further useful functionality for continuation. The toolbox layer contains wrappers to the core that encode algorithms for solving specific classes of continuation problems as well as auxiliary toolboxes that provide additional functionality for these specific classes. As an example, a toolbox for computing and characterizing branches of periodic solutions of ordinary differential equations (ODEs) might make use of an auxiliary toolbox implementing a collocation method for two-point, boundary-value problems of ODEs. Finally, the outermost user layer of an implementation of a continuation problem contains user-provided functions and data that define a specific continuation problem, e.g., the continuation of periodic solutions of a given ODE. The objective of this paper is to propose a novel core design. Compared to existing formulations, the proposed core allows greater flexibility to toolbox developers and more clearly distinguishes between the choices made in deploying a particular covering algorithm and the choices made in formulating a continuation problem. A central theme of the proposed design is the philosophy of an extended continuation problem, a mathematical formulation that naturally supports the idea of task embedding. Using a prototype for a continuation problem with arbitrarily large sets of additional algebraic constraints, namely, the continuation of a hybrid periodic orbit, this paper demonstrates that the extended formulation enables innovative computations that are not supported in a similar way as 'built-in' functionality by any existing core implementations. As argued below, other computations that would profit heavily from support for embedding are the continuation of connecting orbits using algorithms based on Lin's method [4], the computation of Arnol'd tongue scenarios A significant number of computational tools for continuation and bifurcation analysis of characteristic classes of solutions of dynamical systems have been developed in the past and have significantly guided the development effort presented in this paper. These include general algebraic and two-point boundary-value solvers for ordinary differential equations, such as AUTO The remainder of the paper is organized as follows. Section 2 presents the general continuation framework and highlights two common situations that motivate the proposed core design. A mathematical formulation of the extended continuation problem and its advantages is described in Secs. 3 and 4. The reference implementation of the design philosophy in the package COCO and its auxiliary toolboxes is detailed in Sec. 5. Section 6 presents an illustration of the application of the design philosophy and the COCO package to the constrained continuation of periodic trajectories in a hybrid dynamical system modeling a mechanical system with impacts and friction. Finally, a concluding discussion that points the way to further redesign at the toolbox level is presented in Sec. 7

Sensitivity analysis for periodic orbits and quasiperiodic invariant tori using the adjoint method

This is the author accepted manuscript. The final version is available from the American Institute of Mathematical Sciences via the DOI in this recordCode availability: The code included in this paper constitutes fully executable scripts. Complete code, including that used to generate the results in Fig. 1, is available at https://github.com/jansieber/adjoint-sensitivity2022-supp.This paper presents a rigorous framework for the continuation of solutions to nonlinear constraints and the simultaneous analysis of the sensitivities of test functions to constraint violations at each solution point using an adjoint-based approach. By the linearity of a problem Lagrangian in the associated Lagrange multipliers, the formalism is shown to be directly amenable to analysis using the COCO software package, specifically its paradigm for staged problem construction. The general theory is illustrated in the context of algebraic equations and boundary-value problems, with emphasis on periodic orbits in smooth and hybrid dynamical systems, and quasiperiodic invariant tori of flows. In the latter case, normal hyperbolicity is used to prove the existence of continuous solutions to the adjoint conditions associated with the sensitivities of the orbital periods to parameter perturbations and constraint violations, even though the linearization of the governing boundary-value problem lacks a bounded inverse, as required by the general theory. An assumption of transversal stability then implies that these solutions predict the asymptotic phases of trajectories based at initial conditions perturbed away from the torus. Example COCO code is used to illustrate the minimal additional investment in setup costs required to append sensitivity analysis to regular parameter continuation.Engineering and Physical Sciences Research Council (EPSRC

Prediction of stable walking for a toy that cannot stand

Previous experiments [M. J. Coleman and A. Ruina, Phys. Rev. Lett. 80, 3658 (1998)] showed that a gravity-powered toy with no control and which has no statically stable near-standing configurations can walk stably. We show here that a simple rigid-body statically-unstable mathematical model based loosely on the physical toy can predict stable limit-cycle walking motions. These calculations add to the repertoire of rigid-body mechanism behaviors as well as further implicating passive-dynamics as a possible contributor to stability of animal motions.Comment: Note: only corrections so far have been fixing typo's in these comments. 3 pages, 2 eps figures, uses epsf.tex, revtex.sty, amsfonts.sty, aps.sty, aps10.sty, prabib.sty; Accepted for publication in Phys. Rev. E. 4/9/2001 ; information about Andy Ruina's lab (including Coleman's, Garcia's and Ruina's other publications and associated video clips) can be found at: http://www.tam.cornell.edu/~ruina/hplab/index.html and more about Georg Bock's Simulation Group with whom Katja Mombaur is affiliated can be found at http://www.iwr.uni-heidelberg.de/~agboc

Homoclinic crossing in open systems: Chaos in periodically perturbed monopole plus quadrupolelike potentials

The Melnikov method is applied to periodically perturbed open systems modeled by an inverse--square--law attraction center plus a quadrupolelike term. A compactification approach that regularizes periodic orbits at infinity is introduced. The (modified) Smale-Birkhoff homoclinic theorem is used to study transversal homoclinic intersections. A larger class of open systems with degenerated (nonhyperbolic) unstable periodic orbits after regularization is also briefly considered.Comment: 19 pages, 15 figures, Revtex

Energy levels of periodic solutions of the circular 2+2 Sitnikov problem

We study a 2+2 body problem introduced in a previous paper as the circular double Sitnikov problem. Since the secondary bodies are moving on the same perpendicular line where evolve the primaries, almost every solution is a collision orbit. We extend the solutions beyond collisions with a symplectic regularization and study the set of energy surfaces that contain periodic orbits and their foliations .Comment: 20 pages, 5 figures. This is not the final version

On stochastic sea of the standard map

Consider a generic one-parameter unfolding of a homoclinic tangency of an area preserving surface diffeomorphism. We show that for many parameters (residual subset in an open set approaching the critical value) the corresponding diffeomorphism has a transitive invariant set $\Omega$ of full Hausdorff dimension. The set $\Omega$ is a topological limit of hyperbolic sets and is accumulated by elliptic islands. As an application we prove that stochastic sea of the standard map has full Hausdorff dimension for sufficiently large topologically generic parameters.Comment: 36 pages, 5 figure