Intersect transversely

ThenWs(gfe( j) and Wu 7e(t)) intersect transversely at(xh(rio)+0(e),9 j and consequently (from, theS male-Birkhoff homoclinic theorem) for the map P there exists an integer n > 1 that P has an invariant Cantor set on which it is topologically conjugate to a full shift of N symbols. 

Figure 19. (a) Two curves do not intersect transversally, (b) Two curves intersect transversally. 

Figure 3. Illustrating the topological confinement of the orbit in the 4D phase space. The continuous curves T and Y" represent two sets of 2D invariant tori that intersect transversally an energy surface. An orbit with initial datum in the gap between two tori will be eternally trapped in the same region (see text).

On M-y there are locally stable and unstable manifolds that are of equal dimensions and are close to the impertm-bed locally stable and unstable manifolds. The perturbed normally hyperbolic locally invariant manifold intersects each of the 5D level energy sm-faces in a 3D set of which most of the two-parameter family of 2D nonresonant invariant tori persist by the KAM theorem. The Melnikov integral may be computed to determine if the stable and unstable manifolds of the KAM tori intersect transversely. 

Most of the 2D nonresonant invariant tori T(Pi,P2)) that persist are only slightly deformed on the perturbed normally hyperbolic locally invariant manifold and are KAM tori. In the phase space of the perturbed system 7 > 0 and a = 0, there are invariant tori that are densely filled with winding trajectories that are conditionally periodic with two independent frequencies conditionally-periodic motions of the perturbed system are smooth functions of the perturbation 7. A generahzation of the KAM theorem states that the KAM tori have both stable and unstable manifolds by the invariance of manifolds, b fn order to determine if chaos exists, two measurements are required in order to determine whether or not and VK (T.y) intersect transversely. 

Consequently, the stable W Tj Pi,P2)) and mistable W Tj Pi,P2)) manifolds of the KAM tori T.y(Pi, P2) intersect transversely yielding Smale horseshoes on the appropriate 5D level energy surfaces. This implies multiple transverse intersections and the corresponding existence of chaotic djmamics in the perturbed system 7 > 0 and a = 0. 

The manifold M- a has locally stable and unstable manifolds that are close to the unperturbed locally stable and unstable manifolds and if these manifolds intersect transversely, then the Smale-Birkhoff theorem predicts the existence of horseshoes and their chaotic dynamics in the perturbed dissipative system. A 2D hyperbolic invariant torus Tja(Pi, P2) may be located on by averaging the perturbed dissipative vector field 7 > 0 and a > 0 restricted to M q, over the angular variables Qi and Q2- The averaged equations have a unique stable hyperbolic fixed point (Pi,P2) = (0,0) with two negative eigenvalues provided that the... 

Definition 7.10. We will say that TxqWq andTxoWq intersect transversely... 

It is well-known, that if two surfaces intersect transversely at some point, then any two C -close surfaces must intersect transversely at a nearby point. On the contrary, a non-transverse intersection can be removed (or made transverse) by a small perturbation. 

Theorem 7.11. Let L be a saddle periodic orbits and let P be its homoclinic trajectory along which Wf and intersect transversely. Then, any small neighborhood of L JT contains infinitely many saddle periodic orbits. 

Sec. 3.7), we may claim that since W[ intersects transversely, in any small neighborhood U of the point xo, there is a countable set of smooth pieces of converging to n U. Since intersects transversely, it... 

In the Lorenz model, the saddle value is positive for the parameter values corresponding to the homoclinic butterfly. Therefore, upon splitting the two symmetric homoclinic loops outward, a saddle periodic orbit is born from each loop. Furthermore, the stable manifold of one of the periodic orbits intersects transversely the unstable manifold of the other one, and vice versa. The occurrence of such an intersection leads, in turn, to the existence of a hyperbolic limit set containing transverse homoclinic orbits, infinitely many saddle periodic orbits and so on [1]. In the case of a homoclinic butterfly without symmetry there is also a region in the parameter space for which such a rough limit set exists [1, 141, 149]. However, since this limit set is unstable, it cannot be directly associated with the strange attractor — a mathematical image of dynamical chaos in the Lorenz equation. 

Consider the other hypothetical example. Let a two-dimensional diffeo-morphism at e = 0 have a phase portrait as shown in Fig. 14.3.2. Here, O2 and O3 are stable fixed points, and 0 is a saddle. The unstable set Wq of the saddle-node O intersects transversely the stable manifold Wq ... 

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