Half-stable fixed point

Suppose a two-dimensional system has a stable fixed point. What are all the possible ways it could lose stability as a parameter /z varies The eigenvalues of the Jacobian are the key. If the fixed point is stable, the eigenvalues A, must both lie in the left half-plane Re A < 0. Since the A s satisfy a quadratic equation with real coefficients, there are two possible pictures either the eigenvalues are both real and negative (Figure 8.2.1a) or they are complex conjugates (Figure 8.2.1b). To... 

Each of these systems has a fixed point x — 0 with f x ) = 0. However the stability is different in each case. Figure 2.4.1 shows that (a) is stable and (b) is unstable. Case (c) is a hybrid case we II call half-stable, since the fixed point is attracting from the left and repelling from the right. We therefore indicate this type of fixed point by a half-filled circle. Case (d) is a whole line of fixed points perturbations neither grow nor decay. 

As r approaches 0 from below, the parabola moves u p and the two fixed points move toward each other. When r = 0, the fixed points coalesce into a half-stable fixed point at X = 0 (Figure 3.1.1b). This type of fixed point is extremely delicate—it vanishes as soon as r > 0, and now there are no fixed points at all (Figure 3.1.1c),... 

On a resonant invariant curve, out of the infinite set of r-multiple fixed points, only a finite (even) number survive, half stable and half unstable, as a consequence of the Poincare-Birkhoff fixed point theorem (Arnold and Avez, 1968 Lichtenberg and Lieberman, 1983), as shown schematically in Figure 14b. 

Armed as we are now with the KAM theorem, the Center Manifold theorem, and the Stable Manifold theorem, we can begin to visualize the phase space of reaction dynamics. Returning to our original system (see Uncoupled Reaction Dynamics in Two Degrees of Freedom ), we now realize that the periodic orbit that sews together the half-tori to make up the separatrix is a hyperbolic periodic orbit, and it is not a fixed point of reflection. From our previous visualization of uncoupled phase-space dynamics, we know that the separatrix is completely nontwisted. In the terminology of Poliak and Pechukas, the hyperbolic periodic orbit is a repulsive PODS. ... 

It must be pointed out that the thermodynamic work of adhesion can only be reached in stable equilibrium conditions such as sphere on half-space, or DCB at fixed... 

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