Unstable focus equilibrium state

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

The primary scope of this book will focus on the analysis of the internal bifurcations within the class of systems with simple dynamics, such as Morse-Smale systems. Furthermore, we will restrict our study mostly to bifurcations of codimension-one. The reason for this restriction is that some bifurcations of higher codimension turn out to be boundary bifurcations in many cases, such as when the normal forms for the equilibrium states are three-dimensional. Nevertheless, we will examine some codimension-two cases which are concerned with equilibrium states and the loss of stability of periodic orbits. Meanwhile, let us start our next section with a discussion of some questions related to structurally unstable heteroclinic connections. 

If Z/fc > 0, the origin is an unstable equilibrium state because trajectories starting close to it spiral away as time increases. For the two-dimensional system (9.3.1) the point O is called an unstable complex weak) focus. 

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... 

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. ... 

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.

In this chapter, we describe an algorithm for predicting feasible splits for continuous single-feed RD that is not limited by the number of reactions or components. The method described here uses minimal information to determine the feasibility of reactive columns phase equilibrium between the components in the mixture, a reaction rate model, and feed state specification. This is based on a bifurcation analysis of the fixed points for a co-current flash cascade model. Unstable nodes ( light species ) and stable nodes ( heavy species ) in the flash cascade model are candidate distillate and bottom products, respectively, from a RD column. Therefore, we focus our attention on those splits that are equivalent to the direct and indirect sharp splits in non-RD. One of the products in these sharp splits will be a pure component, an azeotrope, or a kinetic pinch point the other product will be in material balance with the first. 

In the preceding chapters we first considered the primary forces acting on a fluidized particle in a bed in equilibrium, and then the elastic forces between particles that come into play under non-equilibrium conditions. These two effects provide closure for the particle bed model, formulated in terms of the particle-and fluid-phase conservation equations for mass and momentum. Up to now, applications have focused on the stability of the state of homogeneous particle suspension, in particular for gas-fluidized systems for which the condition that particle density is much greater than fluid density enables the particle-phase equations to be decoupled and treated independently. The analysis has involved solely the linearized forms of these equations, and has led to a stability criterion that broadly characterizes fluidized systems according to three manifestations of the fluidized state always stable - the usual case for liquids always unstable - the usual case for gases and transitional behaviour - involving a switch, at a critical fluid flux, from the stable to the unstable condition. This characterization has... 

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