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

Model-free Continuation of Periodic Orbits in Certain Nonlinear Systems Using Continuous-Time Adaptive Control

This paper generalizes recent results by the authors on noninvasive model-reference adaptive control designs for control-based continuation of periodic orbits in periodically excited linear systems with matched uncertainties to a larger class of periodically excited nonlinear systems with matched uncertainties and known structure. A candidate adaptive feedback design is also proposed in the case of scalar problems with unmodeled nonlinearities. In the former case, rigorous analysis shows guaranteed performance bounds for the associated prediction and estimation errors. Together with an assumption of persistent excitation, there follows asymptotic convergence to periodic responses determined uniquely by an a priori unknown periodic reference input and independent of initial conditions, as required by the control-based continuation paradigm. In particular, when the reference input equals the sought periodic response, the steady-state control input vanishes. Identical conclusions follow for the case of scalar dynamics with unmodeled nonlinearities, albeit with slow rates of convergence. Numerical simulations validate the theoretical predictions for individual parameter values. Integration with the software package COCO demonstrate successful continuation along families of stable and unstable periodic orbits with a minimum of parameter tuning. The results expand the envelope of known noninvasive feedback strategies for use in experimental model validation and engineering design

Switching adaptive control of a bioassistive exoskeleton

The effectiveness of existing control designs for bioassistive, exoskeletal devices, especially in highly uncertain working environments, depends on the degree of certainty associated with the overall system model. Of particular concern is the robustness of a control design to large-bandwidth exogenous disturbances, time delays in the sensor and actuator loops, and kinematic and inertial variability across the population of likely users. In this study, we propose an adaptive control framework for robotic exoskeletons that uses a low-pass filter structure in the feedback channel to decouple the estimation loop from the control loop. The design facilitates a significant increase in the rate of estimation and adaptation, without a corresponding loss of robustness. In particular, the control implementation is tolerant of time delays in the control loop and maintains clean control channels even in the presence of measurement noise. Tuning of the filter also allows for shaping the nominal response and enhancing the time-delay margin. Importantly, the proposed formulation is independent of detailed model information. The performance of the proposed architecture is demonstrated in simulation for two basic control scenarios, namely, (i) static positioning, for which the predefined desired joint motions are constant; and (ii) command following, where the desired motions are not known a priori and instead inferred using interaction measurements. We consider, in addition, an operating modality in which the control scheme switches between static positioning and command following to facilitate flexible integration of a human operator in the loop. Here, the transition from static positioning to command following is triggered when either the human–machine interaction force at the wrist or the end-effector velocity exceeds the corresponding critical value. The controller switches from command following back to static positioning when both the interaction force and the velocity fall below the corresponding thresholds. This strategy allows for smooth transition between two phases of operation and provides an alternative to an implementation relying on wearable electromyographic sensors

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

Optimization along Families of Periodic and Quasiperiodic Orbits in Dynamical Systems with Delay

This 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.Comment: preprint, 17 pages, 9 figure

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