Unstable invariant manifold

The basins of attraction of the coexisting CA (strange attractor) and SC are shown in the Fig. 14 for the Poincare crosssection oyf = O.67t(mod27t) in the absence of noise [169]. The value of the maximal Lyapunov exponent for the CA is 0.0449. The presence of the control function effectively doubles the dimension of the phase space (compare (35) and (37)) and changes its geometry. In the extended phase space the attractor is connected to the basin of attraction of the stable limit cycle via an unstable invariant manifold. It is precisely the complexity of the structure of the phase space of the auxiliary Hamiltonian system (37) near the nonhyperbolic attractor that makes it difficult to solve the energy-optimal control problem. 

Figure 4. A schematic portrait of the stable and unstable invariant manifolds and the phase-space flows on (4i(p,q),Pi(p,q)).

This enables us to extract and visualize the stable and unstable invariant manifolds along the reaction coordinate in the phase space, to and from the hyperbolic point of the transition state of a many-body nonlinear system. PJ AJI", Pj, Qi, t) and PJ AJ , Pi, q, t) shown in Figure 2.13 can tell us how the system distributes in the two-dimensional (Pi(p,q), qi(p,q)) and PuQi) spaces while it retains its local, approximate invariant of action Jj (p, q) for a certain locality, AJ = 0.05 and z > 0.5, in the vicinity of... 

The multi-dimensional extension of two-dimensional rough systems is the Morse-Smale systems discussed in Sec. 7.4. The list of limit sets of such a system includes equilibrium states and periodic orbits only furthermore, such systems may only have a finite number of them. Morse-Smale systems do not admit homoclinic trajectories. Homoclinic loops to equilibrium states may not exist here because they are non-rough — the intersection of the stable and unstable invariant manifolds of an equilibrium state along a homoclinic loop cannot be transverse. Rough Poincare homoclinic orbits (homoclinics to periodic orbits) may not exist either because they imply the existence of infinitely many periodic orbits. The Morse-Smale systems have properties similar to two-dimensional ones, and it was presumed (before and in the early sixties) that they are dense in the space of all smooth dynamical systems. The discovery of dynamical chaos destroyed this idealistic picture. 

Axiom 2. All periodic orbits and equilibrium states in G are structurally stable and any intersection of their stable and unstable invariant manifolds is transverse. 

Fig 7.5.1. Saddle periodic orbit in R are distinguished by the topology of the stable and unstable invariant manifolds which may be homeomorphic to a cylinder (left) or a Mobius band (right). 

Let us straighten the local stable and unstable invariant manifolds. Then the system near O takes the form (see Sec. 2.7)... 

Fig. 13.4.10. The orientable homoclinic loop (>4 > 0) to a saddle with the positive saddle value has unstable invariant manifold.

Consider a C -smooth (r > 3) system in a neighborhood of a saddle equilibrium state with m-dimensional stable and n-dimensional unstable invariant manifolds. 

This is similar to Case 1, but with Li(0) >0. As e —> —0, a saddle periodic trajectory shrinks into a stable point O. Upon moving through e = 0, the equilibrium state becomes a saddle-focus it spawns a two-dimensional unstable invariant manifold (i.e. the boundary Ss is dangerous). 

When > 0, the unstable invariant manifold of the periodic trajectory has dimension three. 

In contrast, if the choice of the new regime for the representative point is ambiguously defined, then we can assert that such a boimdary is dynamically indefinite. This occurs if at least two attractors belong to the boundary of the unstable set. It must also contain saddles whose unstable invariant manifolds separate the basins of the attractors. 

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