Saddle Connection Research Articles

Perturbations of a Hamiltonian family of cubic vector fields

This paper is related with the configurations of limit cycles for cubic polynomial vector fields in two variables (χ3).It is an open question to decide whether every limit cycle configuration in χ3 can be obtained by perturbation of a corresponding Hamiltonian configuration of centres and graphs.In this work, by considering perturbations of the Hamiltonian vector field XH = (Hy, − Hx), where H(x, y) = [a(x + h)2 + by2 − 1] [a(x − h)2 + by2 − 1], we make a global analysis of the possible cases.The vector field XH has three centres (C−, C+ and the origin) and two saddles. By means of quadratic perturbations the centres become fine foci where C−and C+ have the same type of stability but opposed to that one of the origin and infinity. Further introducing cubic perturbations changes the stability of C−, C+ and the cycle at infinity and generates limit cycles. Lastly extra linear terms change the stability of the origin and generate another limit cycle.Finally, we analyse the rupture of saddle connection of the Hamiltonian field under perturbation, via Melnikov's integral, in order to complete the study of the global phase portrait and to consider the possibility of new limit cycles emerging from the Hamiltonian graph.

Examples of attracting sets of birkhoff type

We discuss an explicit example of a map of the plane R 2 with a nontrivial attracting set. In particular, we are concerned with the concept of rotation number introduced by Poincare for maps of the circle and its subsequent extension by Birkhoff to maps of the annulus. The use of rotation number allows nontrivial attractors to be distinguished. The map we discuss has an attracting set containing a set of orbits with infinitely many different rotation numbers. We obtain the map by considering an Euler iteration of a family of vector fields originally described by Arnold and find that the resulting Euler map undergoes some bifurcations which are analogous to those of the family of vector fields. Specifically, there are Hopf bifurcations where changes of stability of a fixed point result in the creation of an attracting circle. The circle which grows from the fixed point is then shown to undergo structural changes giving nontrivial attracting sets. This arises from Euler map behaviour for which the corresponding vector field behaviour is a heteroclinic saddle connection. It is possible to give an explicit trapping region for the Euler map which contains the attracting set and to describe some of its properties. Finally, an analogy is drawn with attracting sets which arise for forced oscillators.

The growth rate of trajectories of a quadratic differential

AbstractSupposeqis a holomorphic quadratic differential on a compact Riemann surface of genusg≥ 2. Thenqdefines a metric, flat except at the zeroes. A saddle connection is a geodesic joining two zeroes with no zeroes in its interior. This paper shows the asymptotic growth rate of the number of saddles of length at mostTis at most quadratic inT. An application is given to billiards.

Mel’nikov’s Function for Two-Dimensional Mappings

This paper calculates explicitly the Mel’nikoV function for general perturbations of two-dimensional maps possessing a saddle connection. This result may be considered as the discrete analogue of Ziglin’s calculation of the Mel’nikoV function for time-periodic perturbations of two-dimensional continuous flows.Key words. integrable planar maps, perturbations, homoclinic orbits, Mel'nikoV's function, Poisson summation formula

Heteroclinic Orbits and Chaotic Dynamics in Planar Fluid Flows

An extension of the planar Smale–Birkhoff homoclinic theorem to the case of a heteroclinic saddle connection containing a finite number of fixed points is presented. This extension is used to find chaotic dynamics present in certain time-periodic perturbations of planar fluid models. Specifically, the Kelvin–Stuart cat’s eye flow is studied, a model for a vortex pattern found in shear layers. A flow on the two-torus with Hamiltonian $H_0 = (2\pi )^{ - 1} \sin (2\pi x_1 )\cos (2\pi x_2 )$ is studied, as well as the evolution equations for an elliptical vortex in a three-dimensional strain flow.

A perturbation method for saddle connections and homoclinic bifurcation in duffing's equation

A perturbation method is developed for detecting homoclinic connections for the unstable Periodic solution of Duffing's equation with negative linear restoring term. The technique involves a mixture of averaging and singular perturbation methods. Each saddle connection implies a homoclinic point, and the initial bifurcation value of the forcing amplitude is given by the same formula as that obtained by Melnikov's method. The method has also been extended to include an investigation of double loop and transverse saddle connections. Conditions have also been obtained for certain large-amplitude oscillations which are known to occur for this system. The results are illustrated by numerical results for manifolds and saddle connections.

Explosive route to chaos through a fractal torus in a generalized lotka-volterra model

The behavior of a model that generalizes the Lotka-Volterra problem into three dimensions is presented. The results show the analytic derivation of stability diagrams that describe the system's qualitative features. In particular, we show that for a certain value of the bifurcation parameter the system instantly jumps out of a steady state solution into a chaotic solution that portrays a fractal torus in the three-dimensional phase space. This scenario, is referred to as the explosive route to chaos and is attributed to the non-transversal saddle connection type bifurcation. The stability diagrams also present a region in which the Hopf type bifurcation leads to periodic and chaotic solutions. In addition, the bifurcation diagrams reveal a qualitative similarity to the data obtained in the Texas and Bordeaux experiments on the Belousov-Zhabotinskii chemical reaction. The paper is concluded by showing that the model can be useful for representing dynamics associated with biological and chemical phenomena.

Qualitative analysis of a mathematical model for tissue inflammation dynamics

The non-linear differential eguation $$\dot X = \frac{{rX}}{{I + X}} - \frac{{XY}}{{X + Y}},\dot Y = \alpha (I + \theta X - Y)$$ is studied. It is the main aim of this paper to show the existence of bifurcation of saddle connection type; and to show the creation of limit cycles under certain conditions of the parameters, together with their biological significance.

Modulus of stability for vector fields on 3-manifolds

Let X be a vector field on M 3 which exhibits a saddle connection between a singularity p 1 and a periodic orbit σ 1. We give necessary conditions and also sufficient ones in order to have the finite modulus of stability. They rely heavily upon restrictions on the behaviour of p 1 and σ 1.

Disturbance-generated bifurcations in a simple adaptive system: Simulation evidence

We present complete characterization based on computer simulation of the possible dynamics exhibited by an adaptive control system where the (first order) plant with a single unknown pole has constant disturbance at the input. The adaptive system is taken from the current adaptive control systems literature. Here we show that the system in fact undergoes bifurcations and exhibits rich dynamics. As the constant disturbance is varied, the system undergoes saddle-node bifurcation, (subcritical) Hopf bifurcation, and a saddle connection bifurcation. The last bifurcation means that the system acquires a homoclinic orbit for a specific disturbance. While the first two bifurcations are local, the presence of the homoclinic orbit has the potential, when periodically disturbed, to generate nonlocal complicated behavior. This latter behavior is often referred to as ‘horseshoe chaos’.

A codimension two bifurcation with a third order Picard-Fuchs equation

Using a third order Picard-Fuchs equation we show that a certain two parameter family of planar vectorfields for parameter values in a certain cone has a unique limit cycle, which is born from a Hopf bifurcation and dies in a saddle connection. This removes a superfluous hypothesis in Theorem 3.2, Chapter 13 of S. N. Chow and J. K. Hale (“Methods of Bifurcation Theory,” Springer-Verlag, New York, 1982).

Dynamics days la jolla workshop, San Diego, January 3–5, 1984

The dynamical system arising from the problem of billiards is a classical example where the theory of twist maps can be applied. In the case of an elliptic billiard table, the corresponding twist map is integrable and has a saddle connection between two hyperbolic period two points. Using a discrete analog to the Melnikov method, we are able to show that this saddle connection can be deformed into a transversal heteroclinic connection under certain analytic perturbations of the table. From the formulas that we get, we can show that the splitting of the separatrices is exponentially small as a function of the eccentricity of the original unperturbed elliptic table. In addition, we also include a characterization of the period two periodic points for any billiard table.

Sustained Resonance for a Nonlinear System with Slowly Varying Coefficients

J. Kevorkian [SIAM J. Appl. Math., 20 (1971), pp. 364–373; 26 (1974), pp. 638–669] studied resonance for a spinning reentry vehicle using a model system of ordinary differential equations with slowly varying coefficients. He and L. Lewin [SIAM J. Appl. Math., 35 (1978), pp. 738–754] gave formal multiple-time-scale expansions and numerical results to give a description of a mechanism for capture in sustained resonance. J. Sanders [SIAM J. Math. Anal., 10 (1979), pp. 1220–1243] studied these equations more rigorously using the method of averaging, but still could not prove the existence of sustained resonance. This paper continues the study of these equations using higher order averaging and the Melnikov method and shows rigorously that capture in sustained resonance does take place for some initial conditions. The Melnikov method measures the opening of a saddle connection for a small perturbation in terms of an integral. Since it has usually been applied to perturbations which depend periodically on time, the derivation of the integral for systems with slowly varying coefficients is included.

On a gradient-like integro-differential equation

SynopsisUnder appropriate conditions on b and a function g with 2k +1 simple zeros, the equationhas a maximal compact invariant set Ab,g in C([−1,0]R), consisting of the zeros of g and the one-dimensional unstable manifolds of these zeros. For k =2, it is shown that there may be a saddle connection in the flow on Ab,g for some g. This implies that the zeros of g as elements of the flow on Ab,g cannot be given the natural order of the reals.

Averaging and Chaotic Motions in Forced Oscillations

We review the averaging theorem of Krylov–Bogoliubov, which allows one to establish the local existence of periodic orbits in certain forced oscillation problems. We then consider some global features, in particular the existence of hetero- and homoclinic orbits, and describe a method, originally due to Melnikov, by which such orbits can be detected in the case that the averaged equation possesses a saddle connection. We give a number of examples and discuss the chaotic motions resulting from the presence of such orbits.

Bifurcations of the forced van der Pol oscillator

We discuss the bifurcations of the variational equation of the forced van der Pol oscillator and prove the existence of bifurcations of saddle connection type as postulated by M. L. Cartwright [4] and A. W. Gillies [5].