Stable focus equilibrium state

The phase coexistence observed around the first-order transition in NIPA gels cannot be interpreted by the Flory-Rehner theory because this theory tacitly assumes that the equilibrium state of a gel is always a homogeneous one. Heterogeneous structures such as two-phase coexistence are ruled out from the outset in this theory. Of course, if the observed phase coexistence is a transient phenomenon, it is beyond the thermodynamical theory. However, as will be described below, the result of the detailed experiment strongly indicates that the coexistence of phases is not a transient but rather a stable or metastable equilibrium phenomenon. At any rate, we will focus our attention in this article only on static equilibrium phenomena. 

In examining a crystalline structure as revealed by diffraction experiments it is all too easy to view the crystal as a static entity and focus on what may be broadly termed attractive intermolecular interactions (dipole-dipole, hydrogen bonds, van der Waals etc., as detailed in Section 1.8) and neglect the actual mechanism by which a crystal is formed, i.e. the mechanism by which these interactions act to assemble the crystal from a non-equilibrium state in a super-saturated solution. However, it is very often nucleation phenomena that are ultimately responsible for the observed crystal structure and hence we were careful to draw a distinction between solution self-assembly and crystallisation at the beginning of this chapter. For example paracetamol, when crystallised from acetone solution gives the stable monoclinic crystal form I, but crystallisation from a molten sample in the absence of solvent... 

Figure 11.3 State space for different initial conditions. The equilibrium point Pi ( o ) is a stable focus, P2 ( ) is an unstable saddle point, and P3 ( ) is stable.

Another important characteristic of the gas bubble is its response to a periodic oscillation of the ambient pressure / ,. For large-amplitude oscillations of the pressure, or for an initial condition that is not near a stable equilibrium state for the bubble, the response can be very complicated, including the possibility of chaotic variations in the bubble radius.22 However, such features are outside the realm of simple, analytical solutions of the governing equations, and we focus our attention here on the bubble response to asymptotically small oscillations of the ambient pressure, namely,... 

Such equilibrium state is called a weak focus. It is stable if Li < 0 and unstable if L > 0. 

If the first non-zero Lyapunov value is positive and if all non-critical characteristic exponents (71,...,7n) lie to the left of the imaginary axis in the complex plane, then the equilibrium state is a complex saddle-focus, as shown in Fig. 9.3.2(b). Its stable manifold is and the unstable manifold coincides with the center manifold W, The trajectories lying neither in nor pass nearby the equilibrium state. 

Fig. 9.3.2. Two opposite situations in are depicted. When Lfc < 0 the equilibrium state yet preserves its stability on the stability boundary (a) when Lfc > 0 the stable equilibrium state becomes an unstable focus on and, in a global view, a sauidle-focus whose stable...

Let us examine next the bifurcations of the system (11.5.1) in the multidimensional case. If Li < 0 (Fig. 11.5.4), then when // < 0, the equilibrium state O is stable (rough focus when p < 0, and a weak focus aX p = 0) and it attracts all trajectories in a small neighborhood of the origin. When > 0 the point O becomes a saddle-focus with a two-dimensional unstable manifold and an m-dimensional stable manifold. The edge of the unstable manifold is the stable periodic orbit which now attracts all trajectories, except those in the stable manifold of O. One multiplier of the periodic orbit was calculated in Theorem 11.1, this is po p) = 1 — 47r /a (0) -h o p). To find the others we... 

If Li > 0, the phase portraits are depicted in Fig. 11.5.5. Here, when // < 0, there exists a stable equilibrium state O (a focus) and a saddle periodic orbit whose m-dimensional stable manifold is the boundary of the attraction basin of O. As /i increases, the cycle shrinks towards to O and collapses into it at /i = 0. The equilibrium state O becomes a saddle-focus as soon as p increases through zero. 

The asymptotics of curves Li, L3, L4 come from the analysis of the linearization of system (13.2.14) at the equilibrium states. The existence of the curve 1/2 corresponding to the separatrix loop is necessary for the completeness of the bifurcation puzzle the stable cycle generating from the weak focus on the curve L3 must disappear in the loop (when moving towards Z/2)-... 

As we have seen above, the dynamics near the homoclinic loop to a saddle with real leading eigenvalues is essentially two-dimensional. New phenomena appear when we consider the case of a saddle-focus. Namely, we take a C -smooth (r > 2) system with an equilibrium state O of the saddle-focus saddle-focus (2,1) type (in the notation we introduced in Sec. 2.7). In other words, we assume that the equilibrium state has only one positive characteristic exponent 7 > 0, whereas the other characteristic exponents Ai, A2,..., are with negative real parts. Besides, we also assume that the leading (nearest to the imaginary axis) stable exponents consist of a complex conjugate pair Ai and A2 ... 

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. 

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 the equilibrium state is topologically saddle, condition (C.2.8) distinguishes between the cases of a simple saddle and a saddle-focus. However, when the equilibrium is stable or completely unstable, the presence of complex characteristic roots does not necessarily imply that it is a focus. Indeed, if the nearest to the imaginary axis (i.e. the leading) characteristic root is real, the stable (or completely imstable) equilibrium state is a node independently of what other characteristic roots are. 

In the (a, 6)-parameter plane, find the transition boundary saddle-focus for the origin, and equations for its linear stable and unstable subspaces. Detect the curves in the parameter plane that correspond to the vanishing of the saddle value a of the equilibrium state at the origin. Find where the divergence of the vector field at the saddle-focus vanishes. Plot the curves found in the (a 6)-plane. ... 

Fig. C.6.12. Plot of the x-coordinate of the equilibrium state versus z at e = 0. The symbols Xmim a max and (x) denote, respectively, the maximal, minimal and averaged values of the x-coordinates of the stable limit cycle which bifurcates from a stable focus at AH and terminates in the separatrix loop to the saddle O (see the next figure) at the point H z cz 2.086.

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