Numerical approximation of connecting orbits with asymptotic rate

Summary. In this paper we set up and analyze a numerical method for so called {\bf connecting orbits with asymptotic rate } in parameterized dynamical systems. A connecting orbit with asymptotic rate has its initial value in a given submanifold of the phase space (or its cross product with parameter space) and it converges with an exponential rate to a given orbit, e. g. a steady state or a periodic orbit. It is well known that orbits with asymptotic rate can be used to foliate stable or strong stable manifolds of invariant sets. We show that the problem of determining a connecting orbit with asymptotic rate is well-posed if a certain transversality condition is made and a specific relation between the number of stable dimensions and the number of parameters holds. For the proof we employ the implicit function theorem in spaces of exponentially decaying functions. Using asymptotic boundary conditions we truncate the original problem to a finite interval and show that the error decays exponentially. Typically the asymptotic boundary conditions by themselves are the result of a boundary value problem, e. g. if the limiting orbit is periodic. Thus it is expensive to calculate them in a parameter dependent way during the approximation procedure. To avoid this we develop a boundary corrector method which turns out to be nearly optimal after very few steps. Received April 28, 2000 / Revised version received December 18, 2000 / Published online May 30, 2001     

Point-to-Periodic and Periodic-to-Periodic Connections

In this work we consider computing and continuing connecting orbits in parameter dependent dynamical systems. We give details of algorithms for computing connections between equilibria and periodic orbits, and between periodic orbits. The theoretical foundation for these techniques is given by the seminal work of Beyn in 1994, “On well-posed problems for connecting orbits in dynamical systems”, where a numerical technique is also proposed. Our algorithms consist of splitting the computation of the connection from that of the periodic orbit(s). To set up appropriate boundary conditions, we follow the algorithmic approach used by Demmel, Dieci, and Friedman, for the case of connecting orbits between equilibria, and we construct and exploit the smooth block Schur decomposition of the monodromy matrices associated to the periodic orbits. Numerical examples illustrate the performance of the algorithms. This revised version was published online in July 2006 with corrections to the Cover Date.     

Connections to fixed points and Sil’nikov saddle-focus homoclinic orbits in singularly perturbed systems

We consider a singularly perturbed system depending on two parameters with two (possibly the same) normally hyperbolic center manifolds. We assume that the unperturbed system has an orbit that connects a hyperbolic fixed point on one center manifold to a hyperbolic fixed point on the other. Then we prove some old and new results concerning the persistence of these connecting orbits and apply the results to find examples of systems in dimensions greater than three that possess Sil’nikov saddle-focus homoclinic orbits. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 28–55, January, 2008.     

Heteroclinic Transition Motions in Periodic Perturbations of Conservative Systems with an Application to Forced Rigid Body Dynamics

We consider periodic perturbations of conservative systems. The unperturbed systems are assumed to have two nonhyperbolic equilibria connected by a heteroclinic orbit on each level set of conservative quantities. These equilibria construct two normally hyperbolic invariant manifolds in the unperturbed phase space, and by invariant manifold theory there exist two normally hyperbolic, locally invariant manifolds in the perturbed phase space. We extend Melnikov’s method to give a condition under which the stable and unstable manifolds of these locally invariant manifolds intersect transversely. Moreover, when the locally invariant manifolds consist of nonhyperbolic periodic orbits, we show that there can exist heteroclinic orbits connecting periodic orbits near the unperturbed equilibria on distinct level sets. This behavior can occur even when the two unperturbed equilibria on each level set coincide and have a homoclinic orbit. In addition, it yields transition motions between neighborhoods of very distant periodic orbits, which are similar to Arnold diffusion for three or more degree of freedom Hamiltonian systems possessing a sequence of heteroclinic orbits to invariant tori, if there exists a sequence of heteroclinic orbits connecting periodic orbits successively.We illustrate our theory for rotational motions of a periodically forced rigid body. Numerical computations to support the theoretical results are also given.     

Hopf-zero bifurcation of a delayed predator-prey model with dormancy of predators

In this paper, We investigate Hopf-zero bifurcation with codimension 2 in a delayed predator-prey model with dormancy of predators. First we prove the specific existence condition of the coexistence equilibrium. Then we take the mortality rate and time delay as two bifurcation parameters to find the occurrence condition of Hopf-zero bifurcation in this model. Furthermore, using the Faria and Magalhases normal form method and the center manifold theory, we obtain the third order degenerate normal form with two original parameters. Finally, through theoretical analysis and numerical simulations, we give a bifurcation set and a phase diagram to show the specific relations between the normal form and the original system, and explain the coexistence phenomena of several locally stable states, such as the coexistence of multi-periodic orbits, as well as the coexistence of a locally stable equilibrium and a locally stable periodic orbit.     

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 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.     

Well-posedness of the ferrimagnetic equations

In this paper, we show the existence of global weak solutions of the ferrimagnetic equations on compact Riemannian manifold using the penalty method. We also show the uniqueness of the solution and its well-posedness by the energy estimates method in lower dimensions. In particular, when the space dimension is one, we can prove that the problem is globally well-posed.     

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.     

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.     

Robustly transitive actions of ℝ<Superscript>2</Superscript> on compact three manifolds

In this paper, we define robust transitivity for actions of ℝ² on closed connected orientable manifolds. We prove that if the ambient manifold is three dimensional and the dense orbit of a robustly transitive action is not planar, then the action is defined by an Anosov flow, i.e. its orbits coincide with the orbits of an Anosov flow.