Global bifurcation

The mechanism of these transitions is nontrivial and has been discussed in detail elsewhere Q, 12) it involves the development of a homoclinic tangencv and subsequently of a homoclinic tangle between the stable and unstable manifolds of the saddle-type periodic solution S. This tangle is accompanied by nontrivial dynamics (chaotic transients, large multiplicity of solutions etc.). It is impossible to locate and analyze these phenomena without computing the unstable, saddle-tvpe periodic frequency locked solution as well as its stable and unstable manifolds. It is precisely the interactions of such manifolds that are termed global bifurcations and cause in this case the loss of the quasiperiodic solution. 

WIGGINS, S. Global Bifurcations and Chaos. Springer-Verlag, New York, Berlin, Heidelberg, 1988... 

While the conditions 1,2 can be verified approximately by simulation, proving the condition 3 is very difficult. Note that in many studies of chaotic behavior of a CSTR, only the conditions 1,2 are verified, which does not imply chaotic d3mamics, from a rigorous point of view. Nevertheless, the fulfillment of conditions 1,2, can be enough to assure the long time chaotic behavior i.e. that the chaotic motion is not transitory. From the global bifurcations and catastrophe theory other chaotic behavior can be considered throughout the disappearance of a saddle-node fixed point [10], [19], [26]. 

Glass, L., Guevara, M., Belair, J. and Shrier, A., 1984, Global bifurcations of a periodically forced biological oscillator. Phys. Rev. A 29,1348-1357. 

FIGURE 12 Global bifurcation diagrams for eq. (18) near a wigwam singularity. 

Kevrekidis, I. G. 1988 A numerical study of global bifurcations in chemical dynamics. I. systems with two degrees of freedom. A.I.Ch.E. Jl (In the press.)... 

A very important example of global bifurcations is the succession of period-doubling transitions, leading from a steady-state solution to time-periodic solutions of increasing period and finally to nonperiodic behavior. This phenomenon has been studied extensively for iterative equations of the form11,12... 

Fig. 2. Global bifurcation for equations (21)—(23). (a) Hamiltonian reference system, (b) Stable separ-atrix loop arising from the effect of the dissipative" perturbation I2. (c) Asymptotically stable...

The Lorenz equations have a simple structure and contain two nonlinear terms only. Let us briefly consider the main bifurcations in the system (42) (a more detailed analysis can be found in Ref. 181). We fix the parameters ct = 10, b = and vary the parameter r in this case two global bifurcations take place (see the bifurcation diagram in Fig. 20). For r = 1, a supercritical... 

Fig. 2.7 (a) Temporal variation of the membrane potential V and the intracellular calcium concentration S in the considered simple model of a bursting pancreatic cell, (b) Bifurcation diagram forthe fast subsystem the black square denotes a Hopf bifurcation, the open circles are saddle-node bifurcations, and the filled circle represents a global bifurcation, (c) Trajectory plotted on top of the bifurcation diagram. The null-cline forthe slow subsystem is shown dashed. 

Eq. (18) can have for any N, either zero, two, four,. . . or 2N bifurcation points. All the possible local bifurcation diagrams can be constructed by a method described in [1]. Moreover, it can be proven [JJ that any global bifurcation diagram of Eq. (13) must be similar to one of the local bifurcation diagrams of Eq. (18). 

The Isola and Double Limit varieties do not exist in this case. The Hysteresis variety (a =0) divides the a. space into two regions (a > 0 and a. < 0) corresponding to the two bifurcation diagrams shown in Figures 2.a and 2.b. These two are also the only possible global bifurcation diagrams (0 vs. Da) for Eq. (13) as the Hysteresis variety (B 4) divides the B. space into two regions. 1 1... 

S. Wiggins, Global Bifurcation and Chaos—Analytical Methods, Springer, New York, 1988. 

In [STW], a global bifurcation theorem is used to show the existence of a connected global branch of ordered pairs connecting (m, Ei) to... 

The mathematical analysis of (1) requires some advanced techniques from global bifurcation theory see Holmes (1979) or Section 2.2 of Guckenheimer and Holmes (1983). Our more modest goal is to gain some insight into (1) through numerical simulations. 

In the bifurcation diagram shown in Fig. 85, the plane of control parameters was divided into regions of a qualitatively different character of phase trajectories (the shapes of these trajectories are given in the respective regions) and the lines on which occur sensitive states of the Hopf bifurcation and the saddle bifurcation were marked. The diagram also depicts the line of sensitive states of the global bifurcation the appearance of a cycle from the branches of saddle separatrices. 

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