Heteroclinic orbits

In a moving co-ordinate system, the traveling wave equations typically reduce to a system of parameterized nonlinear ordinary differential equations. The solutions of this system corresponding to pulses and fronts for the original reaction-diffusion equation are called homoclinic and heteroclinic orbits, correspondingly, or just connecting orbits. 

We begin the treatment of Theorem 5.1(iv) by showing that the part of the one-dimensional unstable manifold of contained in 2 is a heteroclinic orbit connecting to Ei-... 

Quasiperiodicity is significant because it is a new type of long-term behavior. Unlike the earlier entries (fixed point, closed orbit, homoclinic and heteroclinic orbits and cycles), quasiperiodicity occurs only on the torus. 

In the phase plane p,q),dL front corresponds to trajectory that connects two steady states of (4.4). Such a trajectory is know as a heteroclinic orbit or a heteroclinic connection. The steady states of (4.4) are given by (p, 0), where F(p) = 0. The phase plane trajectories or orbits of (4.4) are the solutions of... 

V < 2V. The state (1,0) is always a saddle point. To be physically acceptable, a front must always be nonnegative. Consequently, only nonnegative heteroclinic orbits are acceptable. Such orbits can only exist if (0,0) is a stable node. In other words, fronts only exist for v > 2 /z5r. Since there exists a heteroclinic connection or front for each value of v with v > 2 /Dt, this analysis does not yield a unique propagating velocity. In fact, the front velocity v depends on the initial condition, specifically on the tail of the initial condition. 

The nonnegative heteroclinic orbit will persist as v decreases, as long as no bifurcation occurs in the vector field of (5.64a) and (5.64c) and (1/2, 1 /2) remains a saddle poinf and (0, 0) a sfable node, whose eigenvectors lie strictly within the positive quadrant. For w < y, the eigenvalues for the fixed point (0, 0) are... 

We employ a method of numerical continuation which has been earlier developed into a software tool for analysis of spatiotemporal patterns emerging in systems with simultaneous reaction, diffusion and convection. As an example, we take a catalytic cross-flow tubular reactor with first order exothermic reaction kinetics. The analysis begins with determining stability and bifurcations of steady states and periodic oscillations in the corresponding homogeneous system. This information is then used to infer the existence of travelling waves which occur due to reaction and diffusion. We focus on waves with constant velocity and examine in some detail the effects of convection on the fiiont waves which are associated with bistability in the reaction-diffusion system. A numerical method for accurate location and continuation of front and pulse waves via a boundary value problem for homo/heteroclinic orbits is used to determine variation of the front waves with convection velocity and some other system parameters. We find that two different front waves can coexist and move in opposite directions in the reactor. Also, the waves can be reflected and switched on the boundaries which leads to zig-zag spatiotemporal patterns. 

A chaotic flow produces either transverse homocHnic or transverse heterocHnic intersections, and/or is able to stretch and fold material in such a way that it produces what is called a horseshoe map, and/or has positive Liapunov exponents. These definitions are not equivalent to each other, and their interrelations have been discussed by Doherty and Ottino [63]. The time-periodic perturbation of homoclinic and heteroclinic orbits can create chaotic flows. In bounded fluid flows, which are encountered in mixing tanks, the homoclinic and heteroclinic orbits are separate streamlines in an unperturbed system. These streamhnes prevent fluid flux from one region of the domain to the other, thereby severely limiting mixing. These separate streamlines generate stable and unstable manifolds upon perturbation, which in turn dictate the mass and energy transports in the system [64-66]. 

The GMM determines the existence of transverse homoclinic/heteroclinic points that are transverse intersections between the stable and unstable manifolds to any invariant sets of the perturbed system when a homoclinic/heteroclinic orbit exists to a hyperbolic invariant manifold in the unperturbed (undamped a = 0 and unperturbed 7 = 0) system. The unperturbed vector field may be computed by setting the perturbation (wavemaker forcing) parameter 7 = 0 and the dissipation parameter a = 0 in (3.45), and are... 

Bertozzi, A. L. Heteroclinic orbits and chaotic dynamics in planar fluid flows II SIAM.J. Math. Anal. 9m,19(6), 1271-1294. 

This result is due to Palis, who had fotmd that two-dimensional diffeomor-phisms with a heteroclinic orbit at whose points an unstable manifold of one saddle fixed point has a quadratic tangency with a stable manifold of another saddle fixed point can be topologically conjugated locally only if the values of some continuous invariants coincide. These continuous invariants are called moduli. Some other non-rough examples where moduli of topological conju-gacy arise are presented in Sec. 8.3. 

The closure of an unclosed Poisson-stable trajectory whose return times are unbounded for some e > 0 is called a quasiminimal set. A quasiminimal set contains, besides Poisson-stable trajectories which are dense everywhere in it, some other invariant and closed subsets. These may be equilibrium states, periodic orbits, non-resonant invariant tori, other minimal sets, homoclinic and heteroclinic orbits, etc., among which a P-trajectory is wandering. This gives a clue to why the recurrent times of the non-trivial unclosed P-trajectory are unbounded. Furthermore, this also points out that Poisson-stable trajectories of a quasiminimal set, due to their unpredictable behavior in time, are of... 

Structurally unstable homoclinic and heteroclinic orbits. Moduli of topological equivalence... 

However, a similar classification of two-dimensional diffeomorphisms, or of three-dimensional fiows, is not that trivial. Let us illustrate this with an example. Consider a diffeomorphism T which has two saddle fixed points 0 and O2 with the characteristic roots )Ai) < 1 and i > 1 at (z = 1,2). Suppose that Wq and have a quadratic tangency along a heteroclinic orbit as shown in Fig. 8.3.1. The quadratic tangency condition implies that all similar diffeomorphisms form a surface of codimension-one in the space of all diffeomorphisms with a C -norm. 

The separatrix values Ai 2 on the heteroclinic orbits are defined in the same way as in the case of homoclinic loops. Note that both cases Ai > 0, A2 > 0 ... 

In particular, let the dimension of the unstable manifold of Oi be equal to the dimension of the imstable manifold of O2. Besides, let both the stable and unstable leading characteristic exponents at both 0 and O2 be real. Assume also that both heteroclinic orbits F 1 2 enter and leave the saddles along the leading directions. We also assume that the extended unstable manifold of one saddle is transverse to the stable manifold of the other saddle along every orbit Fi 2, and that the extended stable manifold of one saddle is transverse to the unstable manifold of the other saddle along F 1,2 as well. Under... 

The bifurcation diagrams are shown in Figs. 13.7.20-13.7.23. The sepa-ratrix values A and A2 are defined as derivatives of the global maps near the heteroclinic orbits Fi and F2 on the two-dimensional invariant manifold. Note that the other cases of combinations of the signs of Ai,2 and of saddle values can be obtained similarly by a reversal of time and a permutation of the sub-indices 1 and 2 . 

The heteroclinic cycles including the saddles whose unstable manifolds have different dimensions were first studied in [34, 35]. This study mostly focused on systems with complex dynamics. Let us, however, discuss here a case where the dynamics is simple. Let a three-dimensional infinitely smooth system have two equilibrium states 0 and O2 with real characteristic exponents, respectively, 7 > 0 > Ai > A2 and 772 > 1 > 0 > (i.e. the unstable manifold of 0 is onedimensional and the unstable manifold of O2 is two-dimensional). Suppose that the two-dimensional manifolds (Oi) and W 02) have a transverse intersection along a heteroclinic trajectory To (which lies neither in the corresponding strongly stable manifold, nor in the strongly unstable manifold). Suppose also that the one-dimensional unstable separatrix of Oi coincides with the one-dimensional stable separatrix of ( 2j so that a structurally unstable heteroclinic orbit F exists (Fig. 13.7.24). The additional non-degeneracy assumptions here are that the saddle values are non-zero and that the extended unstable manifold of Oi is transverse to the extended stable manifold of O2 at the points of the structurally unstable heteroclinic orbit F. 

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