Bifurcation saddle loop

Fig. 26.1a). At first, multistage ignitions and extinctions occur followed by a relaxation (long period) mode [7]. Oscillations die a few degrees below the ignition temperature at a saddle-loop infinite-period homoclinic orbit bifurcation point. This is an example where both ignition and extinction are oscillatory. 

Fig. 18. (a) Example of a combination of a reaction current nFk(if1DL) (dashed curve) in the absence of an inhibiting species P [Eq. (40a), 0 = 0] and an equilibrium coverage of the species P (solid curve) that admits a Hopf bifurcation, (b) Stationary polarization curve in the presence of E and P for overcritical resistance. The dashed line indicates where the stationary state is unstable under galvanostatic conditions. The horizontal bars display the amplitudes of the oscillations, sn saddle-node bifurcation si saddle-loop bifurcation h Hopf bifurcation. 

From Fig. 18b it is clear that under galvanostatic conditions the limit cycle coexists with a stationary state at high overpotentials. The latter is the only attractor at large current densities. Hence, when the current density is increased above the value of the saddle-loop bifurcation, the potential jumps to a steady state far in the anodic region. Once the system has acquired the anodic steady state, it will stay on this branch as the current density is lowered until the stationary state disappears in a saddle-node bifurcation. 

Fig. 19. Skeleton bifurcation diagram in the U/Rq parameter plane of an HN-NDR system of the reaction-inhibition class (such as the prototype model Eqs. (40a,b)). The solid, dashed and dotted-dashed lines denote the location of saddle-node, Hopf and saddle-loop bifurcations, respectively.

To summarize the typical features of HNDR oscillators, they exhibit oscillatory behavior on a branch with a positive characteristic under galvanostatic as well as potentiostatic conditions when a sufficiently large series resistance is involved. At low current densities, the oscillations characteristically set in through a Hopf bifurcation they are predominantly destroyed by a saddle-loop bifurcation at high current densities and they coexist with a stable stationary state at much more anodic values and hence are associated with a hysteresis. 

In an experiment, not all of the features might be clearly visible. In particular, the increase in period close to the saddle-loop bifurcation often occurs in such a brief interval that it is easily missed. Furthermore, the stationary state at high overpotentials commonly lies at such positive values that undesired side reactions, such as oxidation of the electrode, take place. Thus, often an experimentalist will avoid crossing the saddle-loop bifurcation and only investigate the first branch of the diagram. 

Note that from the figure alone, the saddle-loop bifurcation cannot be distinguished from another infinite period bifurcation, a saddle node with infinite period. ... 

Figure 19. Calculated locations of Hopf (h), saddle-node (sn), and saddle-loop (si) bifurcations in the (a) current density-copper concentration and (b) current density-halide concentration parameter planes using the galvanostatic variant of Eqs. (13a-13c) i.e. (u - jtot- (After Wolf et al.

Figure 23. A calculated two-parameter bifurcation diagram for the formic acid model [Eq. (15)] showing the locations of the saddle-node (solid line), Hopf (dashed line), and saddle-loop bifurcations (dotted-dashed line). All three curves meet in a Takens-Bogdanov point close to the cusp. (Reprinted with permission from P. Strasser, M. Eiswirth and G. Ertl, J. Chem. Phys. 107, 991-1003, 1997. Copyright 1997 American Institute of Physics.)...

Fig. 6. Oscillation period P and amplitude N over mixing time tm bifurcation parameter (schematically). X onset of oscillation as an inifinite period saddle-loop bifurcation.

Fig. 20. Existence diagram of various wave forms in the excitable region of model (5). N no waves, F flat solitary pulses, S rigidly rotating spirals, M meandering spirals, T turbulence. For even larger e a limit cycle forms in a saddle-loop bifurcation and subsequently vanishes in a Hopf bifurcation. Turbulence also exists between these two lines after [114]. ...

What cannot be obtained through local bifurcation analysis however, is that both sides of the one-dimensional unstable manifold of a saddle-type unstable bimodal standing wave connect with the 7C-shift of the standing wave vice versa. This explains the pulsating wave it winds around a homoclinic loop consisting of the bimodal unstable standing waves and their one-dimensional unstable manifolds that connect them with each other. It is remarkable that this connection is a persistent homoclinic loop i.e. it exists for an entire interval in parameter space (131. It is possible to show that such a loop exists, based on the... 

Fig. 8.8. Phase plane representations of the birth (or death) of limit cycles through homoclinic orbit formation. In the sequence (a)-fb)-(c) the system has two stable stationary states (solid circles) and a saddle point. As some parameter is varied, the separatrices of the saddle join together to form a closed loop or homoclinic orbit (b) this loop develops as the parameter is varied further to shed an unstable limit cycle surrounding one of the stationary states. The sequence (d)-(e)-(f) shows the corresponding formation of a stable limit cycle which surrounds an unstable stationary state. (In each sequence, the limit cycle may ultimately shrink on to the stationary state it surrounds—at a Hopf bifurcation point.)...

The control loop affects both the static behavior and the dynamic behavior of the system. Our main objective is to stabilize the unstable saddle-type steady state of the system. In the SISO control law (7.72) we use the steady-state values Yfass = 0.872 and Yrdss = 1.5627 as was done in Figures 7.14(a) to (c). A new bifurcation diagram corresponding to this closed-loop case is constructed in Figure 7.20. 

The key to this bifurcation is the behavior of the unstable manifold of the saddle. Look at the branch of the unstable manifold that leaves the origin to the northeast after it loops around, it either hits the origin (Figure 8.4.3c) or veers off to one side or the other (Figures 8.4.3a, d). 

At p = p3, the system jumps to a new state of uniform precession of the director (UP2) with large reorientation (0 74 ) and slow precession rate. As displayed in Fig. 7, starting from the stable UP2 branch above pa and lowering the excitation intensity, one finds a large and rather complicated hysteretic cycle, which eventually flips back to the UPl solution at pg = 1.09. This part of the UP2 branch consists of alternatively stable and unstable regions exhibiting a series of saddle-node bifurcations. Eventually, this branch connects with the UPS one which makes a loop and connects with the UPl branch. 

Surprisingly, even non-rough systems of codimension one may have infinitely many moduli. Of course, since the models of nonlinear dynamics are explicitly defined dynamical systems with a finite set of parameters, this creates a new obstacle which the classical bifurcation theory has not nm into. Although the case of homoclinic loops of codimension one does not introduce any principal problem, nevertheless codimensions two and higher are much less trivial as, for example, in the case of a homoclinic or heteroclinic cycle including a saddle-focus where the structure of the bifurcation diagrams is directly determined by the specific values of the corresponding moduli. 

In Chap. 12 we will study the global bifurcations of the disappearance of saddle-node equilibrium states and periodic orbits. First, we present a multidimensional analogue of a theorem by Andronov and Leontovich on the birth of a stable limit cycle from the separatrix loop of a saddle-node on the plane. Compared with the original proof in [130], our proof is drastically simplified due to the use of the invariant foliation technique. We also consider the case when a homoclinic loop to the saddle-node equilibrium enters the edge of the node region (non-transverse case). 

The bifurcation of a separatrix loop of a saddle-node was discovered by Andronov and Vitt [14] in their study of the transition phenomena from synchronization to beating modulations in radio-engineering. Specifically, they had studied the periodically forced van der Pol equation... 

Another typical codimension-one bifurcation (left untouched in this book) within the class of Morse-Smale systems includes the so-called saddle-saddle bifurcations, where a non-rough saddle equilibrium state with one zero characteristic exponent (the others lie in both left and right half-planes) coalesces with another saddle having a different topological type. If, in addition, the stable and unstable manifolds of the saddle-saddle point intersect each other transversely along some homoclinic orbits, then as the bifurcating point disappears, saddle periodic orbits are born from the homoclinic loops. If there is only one homoclinic loop, then only one periodic orbit is born from it, and respectively, this bifurcation does not lead the system out of the Morse-Smale class. However, if there are more than one homoclinic loops, a hyperbolic limit set with infinitely many saddle periodic orbits will appear after the saddle-saddle vanishes [135]. 

In Chap. 13 we will consider the bifurcations of a homoclinic loop to a saddle equilibrium state. We start with the two-dimensional case. First of all, we investigate the question of the stability of the separatrix loop in the generic case (non-zero saddle value), as well as in the case of a zero saddle value. Next, we elaborate on the cases of arbitrarily finite codimensions where the so-called Dulac sequence is constructed, which allows one to determine the stability of the loop via the sign of the first non-zero term in this sequence. 

In the case of a non-zero saddle value, we present the classical result by Andronov and Leontovich on the birth of a unique limit cycle at the bifurcation of the separatrix loop. Our proof differs from the original proof in [9] where Andronov and Leontovich essentially used the topology of the plane. However, following Andronov and Leontovich we present our proof under a minimal smoothness requirement (C ). 

The case of zero saddle value was considered by E. A. Leontovich in 1951. Her main result is presented in Sec. 13.3, rephrased in somewhat different terms in the case of codimension n (i.e. when exactly the first (n — 1) terms in the Dulac sequence are zero) not more than n limit cycles can bifurcate from a separatrix loop on the plane moreover, this estimate is sharp. 

In the same section we give the bifurcation diagrams for the codimension two case with a first zero saddle value and a non-zero first separatrix value (the second term of the Dulac sequence) at the bifurcation point. Leontovich s method is based on the construction of a Poincare map, which allows one to consider homoclinic loops on non-orientable two-dimensional surfaces as well, where a small-neighborhood of the separatrix loop may be a Mobius band. Here, we discuss the bifurcation diagrams for both cases. 

The bifurcations of periodic orbits from a homoclinic loop of a multidimensional saddle equilibrium state are considered in Sec. 13.4. First, the conditions for the birth of a stable periodic orbit are found. These conditions stipulate that the unstable manifold of the equilibrium state must be one-dimensional and the saddle value must be negative. In fact, the precise theorem (Theorem 13.6) is a direct generalization of the Andronov-Leontovich theorem to the multi-dimensional case. We emphasize again that in comparison with the original proof due to Shilnikov [130], our proof here requires only the -smoothness of the vector field. 

In Sec. 13.5 we consider the bifurcation of the homoclinic loop of a saddle without any restrictions on the dimensions of its stable and imstable manifolds. We prove a theorem which gives the conditions for the birth of a single periodic orbit from the loop [134], and also formulate (without proof) a theorem on complex dynamics in a neighborhood of a homoclinic loop to a saddle-focus. Here, we show how the non-local center manifold theorem (Chap. 6 of Part I) can be used for simple saddles to reduce our analysis to known results (Theorem 13.6). 

Section 13.6 discusses three main cases of codimension-two bifurcations of a homoclinic loop to a saddle. These cases were selected by Shilnikov in [138]... 

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