Saddle regions stability

Figure 39, Chapter 3. Bifurcation diagrams for the model of the Calvin cycle for selected parameters. All saturation parameters are fixed to specific values, and two parameters are varied. Shown is the number of real parts of eigenvalues larger than zero (color coded), with blank corresponding to the stable region. The stability of the steady state is either lost via a Hopf (HO), or via saddle node (SN) bifurcations, with either two or one eigenvalue crossing the imaginary axis, respectively. Intersections point to complex (quasiperiodic or chaotic) dynamics. See text for details.

Fig. 9. Linear stability diagram illustrating (21). The bifurcation parameter B is plotted against the wave number n. (a) and (b) are regions of complex eigenvalues < > (b) shows region corresponding to an unstable focus (c) region corresponding to a saddle point. The vertical lines indicate the allowed discrete values of n. A = 2, D, = 8 103 D2 = 1.6 10-3.

Holmes 1983) states that when the above transversal homoclinic intersection occurs, that there is a structurally stable invariant Cantor set like the one for the Horseshoe map. It has also been shown by Holmes (1982) that this invariant set contains a countable, dense set of saddles of all periods, an uncountable set of non-periodic trajectories and a dense orbit. If nothing else is clear from the above, it is at least certain that homoclinic bifurcations for maps are accompanied by some very unusual phase portraits. Even if homoclinic bifurcations are not necessarily accompanied by the formation of stable chaotic attractors, they lend themselves to extremely long chaotic like transients before settling down to a periodic motion. Because there are large numbers of saddles present, their stable manifolds divide up the phase plane into tiny stability regions and extreme sensitivity to perturbations is expected. 

Point a lies in the middle of the multiplicity region with three steady states. Therefore, this point is an unstable saddle-type steady state. It is clear from Figures 7.21(b) and (c) that this operating point does not correspond to the maximum gasoline yield d-How to alter the operating conditions so that the FCC unit operates at the maximum gasoline yield has been discussed in the previous sections. As explained there, a simple way to stabilize such unstable steady states is to use a negative feedback proportional... 

The phase plane plot of Figure 2 illustrates the behavior of the concentrations of X and Y within this region. Initial concentrations of X and Y corresponding to a point above the broken line will evolve in time to the limit cycle. The broken line represents the separatrix of the middle unstable steady state which has the stability characteristics of a saddle point. Initial values for X and Y corresponding to a point below the separatrix will evolve to the stable state to the right of the diagram. 

The instability arises and evolves owing to thermodynamic fluctua tion (3.29). Such a fluctuation may cause complete system state decay (see, e.g., region V of unstable saddles in Figure 3.4). Flowever, it may also happen that the arising instability creates a new state of the system to be stabilized in time and space. An example is the formation of the limit (restricted) cycle in a system that involves the exceptional point of the unstable focus type. The orbital stability of such a system means exactly the existence of certain time stabilized variations in the thermody namic parameters (for example, the concentrations of reactants) that are... 

Figure 26. Skeleton bifurcation diagram in the t/-p parameter plane for the model equation (16). Shown are Hopf and saddle-node bifurcations (SUN = saddle-unstable-node bifurcation) as well as the border of the focus-node transition (dashed line) mixed-mode wave forms exist close to the dark region (which marks the region where a fixed point is a ShQ nikov saddle focus). The phase portraits sketch the Unear stability of the fixed point(s). (Reprinted with permission from M. T. M. Koper and P. Gaspard, J. Chem. Phys. 96, 7797, 1992. Copyright 1992, American Institute of Physics.)...

Comparing the bifurcation diagram in Fig. 6.5 and its symmetric counterpart in Fig. 6.3 we note that similar stability regions for stationary states, which are limited by the Hopf (H) and saddle-node (LP) bifurcations, exist in both cases. In fact, they can be obtained from each other by continuation along the parameter S. Moreover, as we will see in Sec. 6.4.3, the corresponding branches are close to the synchronous (those that contain (/ = 0) and to the antisynchronous one (containing ip = 7r). 

Fig. 6.6. Stability regions for the stationary states of coupled systems with detuning. LP and H denotes saddle-node and Hopf bifurcations, respectively. ZH is a codimension 2 Zero-Hopf (or Guckenheimer-Gavrilov) bifurcation point. PD - period-doubling bifurcation curve.

In fact, the unstable UPS branch represents the saddle point (or separatrix) that separates the regions of attraction of the NUP state (or, below p2, the UPl state) from that of the largely-reoriented UP2 state. At this point, it might also be interesting to note that the UPl state represents a stable node at p pi (the relevant stabihty exponents are real and negative). Then, between pi and p2, it changes to a focus (the stability exponents become complex). At p2, the real part of the complex pair of stability exponents passes through zero and then becomes positive. 

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