This paper presents a new boundary-value problem formulation for quantifying uncertainty induced by the presence of small Brownian noise near normally hyperbolic attracting periodic orbits (limit cycles) and quasiperiodic invariant tori of the deterministic dynamical systems obtained in the absence of noise. The formulation uses adjoints to construct a continuous family of transversal hyperplanes that are invariant under the linearized deterministic flow near the limit cycle or quasiperiodic invariant torus. The intersections with each hyperplane of stochastic trajectories that remain near the deterministic cycle or torus over intermediate times may be approximated by a Gaussian distribution whose covariance matrix can be obtained from the solution to the corresponding boundary-value problem. In the case of limit cycles, the analysis improves upon results in the literature through the explicit use of state-space projections, transversality constraints, and symmetry-breaking parameters that ensure uniqueness of the solution despite the lack of hyperbolicity along the limit cycle. These same innovations are then generalized to the case of a quasiperiodic invariant torus of arbitrary dimension. In each case, a closed-form solution to the covariance boundary-value problem is found in terms of a convergent series. The methodology is validated against the results of numerical integration for two examples of stochastically perturbed limit cycles and one example of a stochastically perturbed two-dimensional quasiperiodic invariant torus in R2, R2 \times S1, and R2 \times S1, respectively, for which explicit expressions may be found for the associated covariance functions using the proposed series solutions. Finally, an implementation of the covariance boundary-value problem in the numerical continuation package coco is applied to analyze the small-noise limit near a two-dimensional quasiperiodic invariant torus in a nonlinear deterministic dynamical system in R4 that does not support closed-form analysis. Excellent agreement with numerical evidence from stochastic time integration shows the potential for using deterministic continuation techniques to study the influence of stochastic perturbations for both autonomous and periodically excited deterministic vector fields.

This paper presents a new boundary-value problem formulation for quantifying uncertainty induced by the presence of small Brownian noise near transversally stable periodic orbits (limit cycles) and quasiperiodic invariant tori of the deterministic dynamical systems obtained in the absence of noise. The formulation uses adjoints to construct a continuous family of transversal hyperplanes that are invariant under the linearized deterministic flow near the limit cycle or quasiperiodic invariant torus. The intersections with each hyperplane of stochastic trajectories that remain near the deterministic cycle or torus over intermediate times may be approximated by a Gaussian distribution whose covariance matrix can be obtained from the solution to the corresponding boundary-value problem. In the case of limit cycles, the analysis improves upon results in the literature through the explicit use of state-space projections, transversality constraints, and symmetry-breaking parameters that ensure uniqueness of the solution despite the lack of hyperbolicity along the limit cycle. These same innovations are then generalized to the case of a quasiperiodic invariant torus of arbitrary dimension. In each case, a closed-form solution to the covariance boundary-value problem is found in terms of a convergent series. The methodology is validated against the results of numerical integration for two examples of stochastically perturbed limit cycles and one example of a stochastically perturbed two-dimensional quasiperiodic invariant torus. Finally, an implementation of the covariance boundary-value problem in the numerical continuation package coco is applied to analyze the small-noise limit near a two-dimensional quasiperiodic invariant torus in a nonlinear deterministic dynamical system in ℝ^4 that does not support closed-form analysis.

This paper treats comprehensively the construction of problems from nonlinear dynamics and constrained optimization amenable to parameter continuation techniques and with particular emphasis on multi-segment boundary-value problems with delay. The discussion is grounded in the context of the coco software package and its explicit support for community-driven development. To this end, the paper first formalizes the coco construction paradigm for augmented continuation problems compatible with simultaneous analysis of implicitly defined manifolds of solutions to nonlinear equations and the corresponding adjoint variables associated with optimization of scalar objective functions along such manifolds. The paper uses applications to data assimilation from finite time histories and phase response analysis of periodic orbits to identify a universal paradigm of construction that permits abstraction and generalization. It then details the theoretical framework for a coco-compatible toolbox able to support the analysis of a large family of delay-coupled multi-segment boundary-value problems, including periodic orbits, quasiperiodic orbits, connecting orbits, initial-value problems, and optimal control problems, as illustrated in a suite of numerical examples. The paper aims to present a pedagogical treatment that is accessible to the novice and inspiring to the expert by appealing to the many senses of the applied nonlinear dynamicist. Sprinkled among a systematic discussion of problem construction, graph representations of delay-coupled problems, and vectorized formulas for problem discretization, the paper includes an original derivation using Lagrangian sensitivity analysis of phase response functionals for periodic-orbit problems in abstract Banach spaces, as well as a demonstration of the regularizing benefits of multi-dimensional manifold continuation for near-singular problems analyzed using real-time experimental data.

In this paper, we discuss an original covariance boundary-value problem that captures the effects of white noise on the local behavior near stable periodic orbits of the corresponding deterministic dynamical systems. The methodology relies on suitably constructed adjoint variables to project the dynamics onto locally transversal hyperplanes and ensure the existence of a unique solution. For each such hyperplane, the computed co-variance matrix describes an approximately Gaussian, stationary distribution that highlights directions of particular sensitivity to noise. In contrast to previous formulations, the boundary-value problem analyzed in this paper makes predictions in the original state-space variables rather than in terms of a reduced set of local coordinates for hyperplanes perpendicular to the local vector field. The formulation is compatible with the general form of an augmented continuation problem for the software package COCO; a non-adaptive version has been implemented as a general-purpose constructor for the periodic-orbit toolbox included in the COCO release. We illustrate the efficacy of the proposed formulation and toolbox by analyzing noise-induced behavior near limit cycles in two examples of nonlinear dynamical systems with autonomous drift terms. The analysis presented in this study can be used to design safe operating regimes for systems working in stochastic environments, as well as to design optimum working conditions for systems utilizing noise, such as energy-harvesting applications.

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.

We consider a scheduling and sequencing optimization problem with application to an agricultural seeding operation. The problem is motivated by potential deployment of a recently proposed mobile seed refilling system in large-scale field operations. In this setting, mobile refiller vehicles are used to minimize any interruptions to seeding that result from the need to refill the seed tanks on the planters. A feature of this refilling system is that the two types of vehicles can move together during the seed transfer operation. The optimization task seeks to determine locations where seed transfer should be initiated, as well as an ordering of visits of refiller vehicles to such locations, such that the total downtime is minimized. In this paper, we consider such an optimization task for the case of multiple planters and a single refiller that, in turn, can refill itself at a single, fixed, off-field location. We rely on a stochastic solution strategy for solving this problem by applying the genetic algorithm to a binary integer programming model that represents the decision of whether to initiate refilling or not at a discrete set of possible locations along each planter path. The results are compared with a conventional static refilling scheme (with no mobile refiller), as well as a greedy (but generally suboptimal) algorithm that produces a unique solution to the selection and ordering of rendezvous locations. The computational analysis provides deep and useful insights to this optimization problem.

Solitary wave propagation in nonlinear diatomic (dimer) chains is a very interesting topic of research in the study of nonlinear lattices. Such waves were recently found to be supported by the essentially nonlinear granular lattice and Toda lattice. An interesting aspect of this discovery is attributed to the realization of a spectrum of the mass ratio (the only system parameter governing the dynamics) that supports the propagation of such waves corresponding to the considered interaction potential. The objective of this exposition is to explore solitary wave propagation in the dimer binary collision (BC) model. Interestingly, the dimer BC model supports solitary wave propagation at a discrete spectrum of mass ratios similar to those observed in granular and Toda dimers. Further, we report a qualitative and one-to-one correspondence between the spectrum of the mass ratio corresponding to the dimer BC model and those corresponding to granular and Toda dimer chains.

The current study dwells on the wave propagation in one-dimensional periodic granular dimer (diatomic) chain mounted on linear elastic foundation. We invoke multiple time scales and partition the dynamics of the perturbed dimer chain into slow and fast components. An analytical procedure is developed for estimating primary pulse amplitude evolution resulting in a nonlinear map relating the relative displacement amplitudes of two adjacent beads. The evolution predicted by the method of maps is in good agreement with the numerical simulation of the original system. This work forms a basis for application of the devised methodology to weakly coupled granular dimers which finds practical relevance in designing shock mitigating granular layers.

Falling is the leading cause of both fatal and nonfatal injury in the elderly, often requiring expensive hospitalization and rehabilitation. We study the stability of human balance during stance using inverted single- and double-pendulum models, accounting for physiological reflex delays in the controller. The governing second-order neutral delay differential equation (NDDE) is transformed into an equivalent partial differential equation (PDE) constrained by a boundary condition and then into a system of ordinary differential equations (ODEs) using the Galerkin method. The stability of the ODE system approximates that of the original NDDE system; convergence is achieved by increasing the number of terms used in the Galerkin approximation. We validate our formulation by deriving analytical expressions for the stability margins of the double-pendulum human stance model. Numerical examples demonstrate that proportional–derivative–acceleration (PDA) feedback generally, but not always, results in larger stability margins than proportional–derivative (PD) feedback in the presence of reflex delays.

Adaptive reduced-order methods are explored for simulating continuous vibrating structures. The Galerkin method is used to convert the governing partial differential equation (PDE) into a finite-dimensional system of ordinary differential equations (ODEs) whose solution approximates that of the original PDE. Sparse projections of the approximate ODE solution are then found at each integration time step by applying either the least absolute shrinkage and selection operator (lasso) or the optimal subset selection method. We apply the two projection schemes to the simulation of a vibrating Euler–Bernoulli beam subjected to nonlinear unilateral and bilateral spring forces. The subset selection approach is found to be superior for this application, as it generates a solution with similar sparsity but substantially lower error than the lasso.

Adaptive reduced-order methods are explored for simulating continuous vibrating structures. The Galerkin method is used to convert the governing partial differential equation (PDE) into a finite-dimensional system of ordinary differential equations (ODEs) whose solution approximates that of the original PDE. Sparse projections of the approximate ODE solution are then found at each integration time step by applying either the least absolute shrinkage and selection operator (lasso) or the optimal subset selection method. We apply the two projection schemes to the simulation of a vibrating Euler-Bernoulli beam subjected to nonlinear unilateral and bilateral spring forces. The subset selection approach is found to be superior for this application, as it generates a solution with similar sparsity but substantially lower error than the lasso.

In this exposition we consider the wave dynamics of a one-dimensional periodic granular dimer (diatomic) chain mounted on a damped and an undamped linear elastic foundation (otherwise called the on-site potential). It is very well known that periodic granular dimers support solitary wave propagation (similar to that in the homogeneous granular chains) for a specific discrete set of mass ratios. In this work we present the analytical investigation of the evolution of solitary waves and primary pulses in granular dimers when they are mounted on on-site potential with and without velocity proportional foundation damping. We invoke a methodology based on the multiple time-scale asymptotic analysis and partition the dynamics of the perturbed dimer chain into slow and fast components. The dynamics of the dimer chain in the limit of large mass mismatch (auxiliary chain) mounted on on-site potential and foundation damping is used as the basis for the analysis. A systematic analytical procedure is then developed for the slowly varying response of the beads and in estimating primary pulse amplitude evolution resulting in a nonlinear map relating the relative displacement amplitudes of two adjacent beads. The methodology is applicable for arbitrary mass ratios between the beads. We present several examples to demonstrate the efficacy of the proposed method. It is observed that the amplitude evolution predicted by the described methodology is in good agreement with the numerical simulation of the original system. This work forms a basis for further application of the considered methodology to weakly coupled granular dimers which finds practical relevance in designing shock mitigating granular layers.

Finite-dimensional approximations are developed for retarded delay differential equations (DDEs). The DDE system is equivalently posed as an initial-boundary value problem consisting of hyperbolic partial differential equations (PDEs). By exploiting the equivalence of partial derivatives in space and time, we develop a new PDE representation for the DDEs that is devoid of boundary conditions. The resulting boundary condition-free PDEs are discretized using the Galerkin method with Legendre polynomials as the basis functions, whereupon we obtain a system of ordinary differential equations (ODEs) that is a finite-dimensional approximation of the original DDE system. We present several numerical examples comparing the solution obtained using the approximate ODEs to the direct numerical simulation of the original non-linear DDEs. Stability charts developed using our method are compared to existing results for linear DDEs. The presented results clearly demonstrate that the equivalent boundary condition-free PDE formulation accurately captures the dynamic behaviour of the original DDE system and facilitates the application of control theory developed for systems governed by ODEs.

We present an algorithm for determining the stability of delay differential equations (DDEs) with time-periodic coefficients and time-periodic delays. The DDEs are first posed as an equivalent system of partial differential equations (PDEs) along with a nonlinear boundary condition. A Galerkin approximation is then employed to discretize the PDEs into a set of time-periodic ordinary differential equations (ODEs). Finally, we apply a modified version of the Arnoldi algorithm to extract the dominant eigenvalues of the Floquet transition matrix without computing the entire matrix, thereby reducing the required number of integrations of the ODE system. Five numerical examples demonstrate that our modified Arnoldi algorithm provides reliable approximations of the dominant eigenvalues of the Floquet transition matrix, and does so with substantially less computational effort than the classical Floquet method. The stability charts and bifurcation diagrams generated using our Galerkin–Arnoldi method clearly demonstrate the utility of this approach for establishing the stability of a system of DDEs.

Finite-dimensional approximations are developed for retarded delay differential equations (DDEs). The DDE system is equivalently posed as an initial–boundary value problem consisting of hyperbolic partial differential equations (PDEs). By exploting the equivalence of partial derivatives in space and time, we develop a new PDE representation for the DDEs that is devoid of boundary conditions. The resulting boundary condition–free PDEs are discretized using the Galerkin method with Legendre polynomials as the basis functions, whereupon we obtain a system of ordinary differential equations (ODEs) that is a finite-dimensional approximation of the original DDE system.