Connecting Orbits Research Articles

Computer assisted error bounds for linear approximation of (un)stable manifolds and rigorous validation of higher dimensional transverse connecting orbits

This paper presents a method for computing validated error bounds on the truncation error associated with the linear approximation of an (un)stable invariant manifold by its eigenspace. The method is based on studying a certain functional equation which describes a chart map for the invariant manifold. The truncation error is represented as a bounded analytic function on an explicitly given neighborhood. Moreover studying this functional equation leads to a method for bounding derivatives of the truncation error as well. The methods developed in the present work are well suited for studying higher dimensional invariant manifolds, and validated numerical results are provided for manifolds of dimension up to one hundred. As an application of these ideas we present some computer assisted proofs of transverse homoclinic connecting orbits.

Computer Assisted Proof of Transverse Saddle-to-Saddle Connecting Orbits for First Order Vector Fields

In this paper we introduce a computational method for proving the existence of generic saddle-to-saddle connections between equilibria of first order vector fields. The first step consists of rigorously computing high order parametrizations of the local stable and unstable manifolds. If the local manifolds intersect, the Newton–Kantorovich theorem is applied to validate the existence of a so-called short connecting orbit. If the local manifolds do not intersect, a boundary value problem with boundary values in the local manifolds is rigorously solved by a contraction mapping argument on a ball centered at the numerical solution, yielding the existence of a so-called long connecting orbit. In both cases our argument yields transversality of the corresponding intersection of the manifolds. The method is applied to the Lorenz equations, where a study of a pitchfork bifurcation with saddle-to-saddle stability is done and where several proofs of existence of short and long connections are obtained.

A variational approach to connecting orbits in nonlinear dynamical systems

We propose a variational method for determining homoclinic and heteroclinic orbits including spiral-shaped ones in nonlinear dynamical systems. Starting from a suitable initial curve, a homotopy evolution equation is used to approach a true connecting orbit. The procedure is an extension of a variational method that has been used previously for locating cycles, and avoids the need for linearization in search of simple connecting orbits. Examples of homoclinic and heteroclinic orbits for typical dynamical systems are presented. In particular, several heteroclinic orbits of the steady-state Kuramoto–Sivashinsky equation are found, which display interesting topological structures, closely related to those of the corresponding periodic orbits. • A variational technique for determining connecting orbits is proposed. • It does not need linearization for searching simple connecting orbits. • The initialization and parameter evolution becomes particularly simple. • New connections of the steady-state Kuramoto–Sivashinsky equation are found.

Pattern formation in Rayleigh–Bénard convection

The main objective of this article is to study the three-dimensional Rayleigh-Benard convection in a rectangular domain from a pattern formation perspective. It is well known that as the Rayleigh number crosses a critical threshold, the system undergoes a Type-I transition, characterized by an attractor bifurcation. The bifurcated attractor is an (m-1)-dimensional homological sphere where m is the multiplicity of the first critical eigenvalue. When m=1, the structure of this attractor is trivial. When m=2, it is known that the bifurcated attractor consists of steady states and their connecting heteroclinic orbits. The main focus of this article is then on the pattern selection mechanism and stability of rolls, rectangles and mixed modes (including hexagons) for the case where m=2. We derive in particular a complete classification of all transition scenarios, determining the patterns of the bifurcated steady states, their stabilities and the basin of attraction of the stable ones. The theoretical results lead to interesting physical conclusions, which are in agreement with known experimental results. For example, it is shown in this article that only the pure modes are stable whereas the mixed modes are unstable.

DYNAMIC BIFURCATION OF THE PERIODIC SWIFT-HOHENBERG EQUATION

In this paper we study the dynamic bifurcation of the Swift- Hohenberg equation on a periodic cell = ( L,L). It is shown that the equations bifurcates from the trivial solution to an attractor Awhen the control parametercrosses the critical value. In the odd periodic case, Ais homeomorphic to S1 and consists of eight singular points and their connecting orbits. In the periodic case, Ais homeomorphic to S1, and contains a torus and two circles which consist of singular points. Fluid motion driven by the thermal gradients is common in nature, especially in geophysical flows such as the atmosphere, the oceans, the mantle of the earth, and the interior of stars. A typical model for fluid convection is the Rayleigh- Benard convection describing a fluid placed between flat horizontal plates such that the lower plate is maintained at a temperature above the upper plate temperature. Due to the thermal expansion, the fluid near the lower plate is less dense and become unstable in the gravitational field. Eventually, we encounter an instability at a finite wave length giving a spatio-temporal pattern formation. The mathematical model for the Rayleigh-Benard convection comes from the equation of fluid dynamics in the Boussinesq approximation which involves the Navier-Stokes equations coupled with the temperature equation. In 1977, Swift and Hohenberg derived in (14) that when the Rayleigh number is near the onset of the convection, the Rayleigh-Benard convection model may be approximated by the following Swift-Hohenberg equation (SHE)

Numerical computation of connecting orbits on a manifold

In this paper we propose a numerical method for approximating connecting orbits on a manifold and its bifurcation parameters. First we extend the standard nondegeneracy condition to the connecting orbits on a manifold. Then we construct a well-posed system such that the nondegenerate connecting orbit pair on a manifold is its regular solution. We use a difference method to discretize the ODE part in this well-posed system and we find that the numerical solutions still remain on the same manifold. We also set up a modified projection boundary condition to truncate connecting orbits on a manifold onto a finite interval. Then we prove the existence of truncated approximate connecting orbit pairs and derive error estimates. Finally, we carry out some numerical experiments to illustrate the theoretical estimates.

Phase transition for potentials of high‐dimensional wells

AbstractFor a potential function $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}F:\R^k \to\R _ +$ that attains its global minimum value at two disjoint compact connected submanifolds N± in $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}\R^k$, we discuss the asymptotics, as ϵ → 0, of minimizers uϵ of the singular perturbed functional ${\bf E}_\varepsilon (u) = \int_\Omega {(|\nabla u|^2 + {1 \over {\varepsilon ^2 }}F(u))} dx$ under suitable Dirichlet boundary data $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}g_\varepsilon :\partial \Omega \to\R ^k$. In the expansion of Eϵ (uϵ) with respect to ${1 \over \varepsilon }$, we identify the first‐order term by the area of the sharp interface between the two phases, an area‐minimizing hypersurface Γ, and the energy c of minimal connecting orbits between N+ and N−, and the zeroth‐order term by the energy of minimizing harmonic maps into N± both under the Dirichlet boundary condition on ∂Ω and a very interesting partially constrained boundary condition on the sharp interface Γ. © 2012 Wiley Periodicals, Inc.

Dynamical bifurcation of the two dimensional Swift-Hohenberg equation with odd periodic condition

In this article, we study the stability and dynamic bifurcation for the two dimensional Swift-Hohenberg equation with an odd periodic condition. It is shown that an attractor bifurcates from the trivial solution as the control parameter crosses the critical value. The bifurcated attractor consists of finite number of singular points and their connecting orbits. Using the center manifold theory, we verify the nondegeneracy and the stability of the singular points.

A new method to find homoclinic and heteroclinic orbits

A new series method is provided for continuous-time autonomous dynamical systems, which can find exact orbits as opposed to approximate ones. The method can reduce the connecting orbit problem as a boundary value problem in an infinite time domain to the initial value problem. It consists of transforming time to the logarithmic scale, substituting a power series around each fixed point of interest for each of the unknown functions into the system, and equating the corresponding coefficients. When solving for the power series coefficients, additional parameters are used in order to find the intersections of the unstable manifold and the stable manifold of the equilibria. This paper demonstrates how the new method allows to obtain heteroclinic and homoclinic orbits in some well-known cases, such as Nagumo system, stretch-twist-fold flow or mathematical pendulum.

Rigorous Numerics for Symmetric Connecting Orbits: Even Homoclinics of the Gray–Scott Equation

In this paper we propose a rigorous numerical technique for the computation of symmetric connecting orbits for ordinary differential equations. The idea is to solve a projected boundary value problem (BVP) in a function space via a fixed point argument. The formulation of the projected BVP involves a high order parameterization of the invariant manifolds at the steady states. Using this parameterization, one can obtain explicit exponential asymptotic bounds for the coefficients of the expansion of the manifolds. Combining these bounds with piecewise linear approximations, one can construct a contraction in a function space whose unique fixed point corresponds to the wanted connecting orbit. We have implemented the method to demonstrate its effectiveness, and we have used it to prove the existence of a family of even homoclinic orbits for the Gray–Scott equation.

Investigating the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields

We consider a homoclinic bifurcation of a vector field in $\R^3$, where a one-dimensional unstable manifold of an equilibrium is contained in the two-dimensional stable manifold of this same equilibrium. How such one-dimensional connecting orbits arise is well understood, and software packages exist to detect and follow them in parameters. &nbsp&nbsp In this paper we address an issue that it is far less well understood: how does the associated two-dimensional stable manifold change geometrically during the given homoclinic bifurcation? This question can be answered with the help of advanced numerical techniques. More specifically, we compute two-dimensional manifolds, and their one-dimensional intersection curves with a suitable cross-section, via the numerical continuation of orbit segments as solutions of a boundary value problem. In this way, we are able to explain how homoclinic bifurcations may lead to quite dramatic changes of the overall dynamics. This is demonstrated with two examples. We first consider a Shilnikov bifurcation in a semiconductor laser model, and show how the associated change of the two-dimensional stable manifold results in the creation of a new basin of attraction. We then investigate how the basins of the two symmetrically related attracting equilibria change to give rise to preturbulence in the first homoclinic explosion of the Lorenz system.

CONTINUATION OF CONNECTING ORBITS IN 3D-ODEs (II) : CYCLE-TO-CYCLE CONNECTIONS

In Part I of this paper we have discussed new methods for the numerical continuation of point-to-cycle connecting orbits in three-dimensional autonomous ODE's using projection boundary conditions. In this second part we extend the method to the numerical continuation of cycle-to-cycle connecting orbits. In our approach, the projection boundary conditions near the cycles are formulated using eigenfunctions of the associated adjoint variational equations, avoiding costly and numerically unstable computations of the monodromy matrices. The equations for the eigenfunctions are included in the defining boundary-value problem, allowing a straightforward implementation in AUTO, in which only the standard features of the software are employed. Homotopy methods to find the connecting orbits are discussed in general and illustrated with an example from population dynamics. Complete AUTO demos, which can be easily adapted to any autonomous three-dimensional ODE system, are freely available.

CONTINUATION OF CONNECTING ORBITS IN 3D-ODEs (I): POINT-TO-CYCLE CONNECTIONS

We propose new methods for the numerical continuation of point-to-cycle connecting orbits in three-dimensional autonomous ODE's using projection boundary conditions. In our approach, the projection boundary conditions near the cycle are formulated using an eigenfunction of the associated adjoint variational equation, avoiding costly and numerically unstable computations of the monodromy matrix. The equations for the eigenfunction are included in the defining boundary-value problem, allowing a straightforward implementation in AUTO, in which only the standard features of the software are employed. Homotopy methods to find connecting orbits are discussed in general and illustrated with several examples, including the Lorenz equations. Complete AUTO demos, which can be easily adapted to any autonomous three-dimensional ODE system, are freely available.

A Lin's method approach to finding and continuing heteroclinic connections involving periodic orbits

We present a numerical method for finding and continuing heteroclinic connections of vector fields that involve periodic orbits. Specifically, we concentrate on the case of a codimension-d heteroclinic connection from a saddle equilibrium to a saddle periodic orbit, denoted EtoP connection for short. By employing a Lin's method approach we construct a boundary value problem that has as its solution two orbit segments, one from the equilibrium to a suitable section Σ and the other from Σ to the periodic orbit. The difference between their two end points in Σ can be chosen in a d-dimensional subspace, and this gives rise to d well-defined test functions that are called the Lin gaps. A connecting orbit can be found in a systematic way by closing the Lin gaps one by one in d consecutive continuation runs. Indeed, any common zero of the Lin gaps corresponds to an EtoP connection, which can then be continued in system parameters.The performance of our method is demonstrated with a number of examples. Specifically, we computate different types of EtoP orbits in the Lorenz system, in a vector-field model of a saddle-node Hopf bifurcation with global reinjection and in a four-dimensional Duffing-type system. Finally, we demonstrate the versatility of our geometric approach by finding a codimension-zero heteroclinic connection between two saddle periodic orbits in a four-dimensional vector field.

Transversal connecting orbits from shadowing

A rigorous numerical method for establishing the existence of a transversal connecting orbit from one hyperbolic periodic orbit to another of a differential equation in $${\mathbb{R}^n}$$ is presented. As the first component of this method, a general shadowing theorem that guarantees the existence of such a connecting orbit near a suitable pseudo connection orbit given the invertibility of a certain operator is proved. The second component consists of a refinement procedure for numerically computing a pseudo connecting orbit between two pseudo periodic orbits with sufficiently small local errors so as to satisfy the hypothesis of the theorem. The third component consists of a numerical procedure to verify the invertibility of the operator and obtain a rigorous upper bound for the norm of its inverse. Using this method, existence of chaos is demonstrated on examples with transversal homoclinic orbits, and with cycles of transversal heteroclinic orbits.

Travelling waves for delayed reaction–diffusion equations with global response

We develop a new approach to obtain the existence of travelling wave solutions for reaction–diffusion equations with delayed non-local response. The approach is based on an abstract formulation of the wave profile as a solution of an operational equation in a certain Banach space, coupled with an index formula of the associated Fredholm operator and some careful estimation of the nonlinear perturbation. The general result relates the existence of travelling wave solutions to the existence of heteroclinic connecting orbits of a corresponding functional differential equation, and this result is illustrated by an application to a model describing the population growth when the species has two age classes and the diffusion of the individual during the maturation process leads to an interesting non-local and delayed response for the matured population.

NUMERICAL ANALYSIS OF DEGENERATE CONNECTING ORBITS FOR MAPS

This paper contains a survey of numerical methods for connecting orbits in discrete dynamical systems. Special emphasis is put on degenerate cases where either the orbit loses transversality or one of its endpoints loses hyperbolicity. Numerical methods that approximate the connecting orbits by finite orbit sequences are described in detail and theoretical results on the error analysis are provided. For most of the degenerate cases we present examples and numerical results that illustrate the applicability of the methods and the validity of the error estimates.

On connecting orbits for competing species equations with large interactions

We use homotopy index and monotonicity techniques to study the connecting orbits of systems of two competing species equations with diffusion and large interaction. We also use earlier work of Zhitao Zhang and the author on the dynamics of this system.

A Two-Parameter Study of the Locking Region of a Semiconductor Laser Subject to Phase-Conjugate Feedback

We present a detailed bifurcation analysis of a single-mode semiconductor laser subject to phase-conjugate feedback, a system described by a delay differential equation. Codimension-one bifurcation curves of equilibria and periodic orbits and curves of certain connecting orbits are presented near the laser's locking region in the two-dimensional parameter plane of feedback strength and pump current. We identify several codimension-two bifurcations, including a double-Hopf point, Belyakov points, and a T-point bifurcation, and we show how they organize the dynamics. This study is the first example of a two-parameter bifurcation study, including bifurcations of periodic and connecting orbits, of a delay system. It was made possible by new numerical continuation tools, implemented in the package DDE-BIFTOOL, and showcases their usefulness for the study of delay systems arising in applications.

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.