Orbits In Delay Differential Equations Research Articles

Stability and Bifurcations in Time-Delay Systems: A Novel and Efficient Blend of Methods

The goal of this paper is to introduce a hybrid technique, combining two existing methods conveniently, to analyze the stability and bifurcations of equilibrium points and periodic orbits in delay-differential equations of retarded type. One method is called the frequency-domain approach, an analytical tool based on control theory allowing to detect and represent periodic solutions emerging from the Hopf bifurcations, via Fourier series and harmonic balances. The second one, the so-called semi-discretization method, provides a numerical scheme to approximate the monodromy operator of a periodic solution, thus permitting to establish the stability and bifurcations of limit cycles as well as analyzing the equilibrium points. It is shown that a proper combination of these methods provides a straightforward strategy to study both local and global bifurcations in time-delay systems. The usefulness of the proposed approach is shown by three examples, where the results are compared with those given by the software DDE-BIFTOOL.

Self-induced synchronization by large delay

We investigate the dynamics of a delay differential equation obtained by perturbing a vector field f:Rn→Rn, admitting a stable periodic orbit, by using a delayed feedback control term ηg(x,x(t−τ)) of same regularity, where η is small and τ large so that ητ be bounded but non small. We prove that trajectories starting in a neighborhood (of size independent of the parameters η,τ) of this original periodic orbit enter asymptotically a periodic regime, and that the number of such distinct periodic regimes increases (almost) linearly when ητ increases infinitely. Our result is based on the construction of an invariant manifold via a process inspired by the Lyapunov-Perron method for integral operators associated to ordinary differential equations, and on the persistence of normally hyperbolic invariant manifolds for semi-flows on Banach spaces. The statement we provide here complements already known results on periodic orbits of delay differential equations.

Using automatic differentiation to compute periodic orbits of delay differential equations

<p style='text-indent:20px;'>In this paper we focus on the computation of periodic solutions of Delay Differential Equations (DDEs) with constant delays. The method is based on defining a Poincaré section in a suitable functional space and looking for a fixed point of the flow in this section. This is done by applying a Newton method on a suitable discretisation of the section. To avoid computing and storing large matrices we use a GMRES method to solve the linear system because in this case GMRES converges very fast due to the compactness of the flow of the DDE. The derivatives of the Poincaré map are obtained in a simple way, by applying Automatic Differentiation to the numerical integration. The stability of the periodic orbit is also obtained in a very efficient way by means of Arnoldi methods. The examples considered include temporal and spatial Poincaré sections.

Continuation of periodic solutions of various types of delay differential equations using asymptotic numerical method and harmonic balance method

This article presents an extension of the asymptotic numerical method combined with the harmonic balance method to the continuation of periodic orbits of delay differential equations. The equations can be forced or autonomous and possibly of neutral type. The approach developed in this paper requires the system of equations to be written in a quadratic formalism which is detailed. The method is applied to various systems, from Van der Pol and Duffing oscillators to toy models of clarinet and saxophone. The harmonic balance method is ascertained from a comparison to standards time-integration solvers. Bifurcation diagrams are drawn which are sometimes intricate, showing the robustness of this method.

Search for Periodic Orbits in Delay Differential Equations

In a previous paper, we developed a new way to apply the Homotopy Analysis Method (HAM) in the search for periodic orbits in dynamical systems modeled by ordinary differential equations. This method differs from the original in the heuristic search of the frequencies of the cycles. In this paper, we show that the method can be extended to the search for periodic orbits in delay differential equations. Herein, this methodology is applied twice, firstly in an equation of van der Pol type and secondly in an anharmonic oscillator, both systems with a delayed feedback.

Computing unstable manifolds of periodic orbits in delay differential equations

We present the first algorithm for computing unstable manifolds of saddle-type periodic orbits with one unstable Floquet multiplier in systems of autonomous delay differential equations (DDEs) with one fixed delay. Specifically, we grow the one-dimensional unstable manifold W u( q) of an associated saddle fixed point q of a Poincaré map defined by a suitable Poincaré section Σ. Starting close to q along the linear approximation to W u( q) given by the associated eigenfunction, our algorithm grows the manifold as a sequence of points, where the distance between points is governed by the curvature of the one-dimensional intersection curve W u( q)∩ Σ of W u( q) with Σ. Our algorithm makes it possible to study global bifurcations in DDEs. We illustrate this with the break-up of an invariant torus and a subsequent crisis bifurcation to chaos in a DDE model of a semiconductor laser with phase-conjugate feedback.

Discretizing hyperbolic periodic orbits of delay differential equations

In this paper we generalize the results on discretizing exponentially stable periodic orbits of delay differential equations obtained in [9] to the case of unstable hyperbolic periodic orbits. Moreover, we prove persistence of local invariant manifolds around the periodic orbit and a shadowing theorem which allows us to compare the dynamics of the original solution operator and its discretization in a neighborhood of the periodic orbit. The proofs use ideas from [1] but our assumptions on the closeness of the solution operator and its discretization — motivated by concrete numerical applications — are weaker. To overcome the difficulty caused by this weaker closeness assumption we use the smoothing effect of delay equations.

Numerical computation of connecting orbits in delay differential equations

We discuss the numerical computation of homoclinic and heteroclinic orbits in delay differential equations. Such connecting orbits are approximated using projection boundary conditions, which involve the stable and unstable manifolds of a steady state solution. The stable manifold of a steady state solution of a delay differential equation (DDE) is infinite-dimensional, a problem which we circumvent by reformulating the end conditions using a special bilinear form. The resulting boundary value problem is solved using a collocation method. We demonstrate results, showing homoclinic orbits in a model for neural activity and travelling wave solutions to the delayed Hodgkin–Huxley equation. Our numerical tests indicate convergence behaviour that corresponds to known theoretical results for ODEs and periodic boundary value problems for DDEs.

Periodic orbits of delay differential equations under discretization

This paper deals with the long-time behaviour of numerical solutions of delay differential equations that have asymptotically stable periodic orbits. It is shown that Runge-Kutta discretizations of such equations have attractive invariant curves which approximate the periodic orbit with the order of the method.

Melnikov functions and heteroclinic orbits in delay differential equations

Let C : = C[ -r, 01, r E [0, + co), be the Banach space of continuous functions 4 : [ - r, 0] + R” with the sup norm. For any continuous function z:R-tR” and PER, Z~ E C is defined by means of z,(0) = z(t+ e), -r < 8 0. The main aim of this paper is to give sufficient conditions assuring the existence of bounded solutions to perturbed delay differential equations (D.D.E.) of the following type: