Multi-stability

The entropy export to the environment becomes possible if the system is provided continuously with an overcritical amount of the energy. So the system can leave equilibrium or does never reach it. Multi-stability can occur in different types of systems. Some of the typical examples are given in Table 8.2. 

Animal psychology Bistable behaviour of an animal under stress (flight or attack) 

Economics Two stable levels of good production (a) Low production with high prices (b) High production with low prices 

Nerve physiology (a) Switching mechanism in nerve cells from rest to achon potential (b) Epileptic brain, seizure in epilepsy 

Biochemistry Bistable mechanism in enzyme reaction, protein/biochemical systems 

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Our understanding of the development of oscillations, multi-stability and chaos in well stirred chemical systems and pattern fonnation in spatially distributed systems has increased significantly since the early observations of these phenomena. Most of this development has taken place relatively recently, largely driven by development of experimental probes of the dynamics of such systems. In spite of this progress our knowledge of these systems is still rather limited, especially for spatially distributed systems. 

Next, consider the case with p = 0.02014. The traverse across Fig. 12.6(a) as r is varied now also cuts the region of multi stability. It passes above the cusp point C (see Fig. 12.5), giving rise to two turning points in the stationary-state locus, but below the double-zero eigenvalue point M. There are still four intersections with the Hopf curve, so there are four points of Hopf bifurcation. The Hopf point at highest r is now a subcritical bifurcation. The dependence of the reaction rate on r for this system is shown in Fig. 12.6(d). 

Chapter 8. Bifurcation Phenomenon and Multi-stability which can be simplified to the model reaction system... 

Equilibrium state —Linear steady state close to equilibrium —Steady state —> Non-linear steady state — Bifurcation phenomena —> Multi-stability —> Temporal and spatio-temporal oscillations —> More complex situations (chaos, turbulence, pattern formation, fractal growth). All these stages have been discussed in different chapters of the book. 

One might well ask whether the advent of new types of oscillators has brought with it the discovery of new dynamical phenomena. Of course, the BZ reaction itself exhibits an enormous variety of behavior including chaos, spatial waves, multi stability and complex oscillation. Therefore there may not be many new phenomena left to discover (this of course, is always a dangerous view to take). 

A simple two-variable theoretical model is proposed. It exhibits most of the temporal dynamical behaviours reported for nonlinear chemical systems [1] oscillations, excitability, multi stability. This autocatalytic model, part of our nonlinear model of calcium metabolism [2], has been associated with bone calcification processes (nucleation and crystal growth). It is described by the differential system ... 

The prototype, cubic autocatalytic reaction (A + 2B 3B) forms the basis of a simple homogeneous system displaying a rich variety of complex behaviour. Even under well-stirred, isothermal open conditions (the CSTR) we may find multi stability, hysteresis, extinction and ignition. Allowing for the finite lifetime of the catalyst (B inert products) adds another dimension. The dependence of the stationary-states on residence-time now yields isolas and mushrooms. Sustained oscillations (stable limit cycles) are also possible. There are strong analogies between this simple system and the exothermic, first-order reaction in a CSTR. 

Next we perform numerical experiments with the Brusselator in a disc inscribed in a square, i.e., of diameter L. As in the sinusoidal case, there is no multi-stability of trajectories, and in particular the central position of the circle is an unstable position for the defect. There is only one stable trajectory along the circular boundaries, but contrary to the sinusoidal oscillations, there is only a translational motion and no looping motion is seen. 

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