Saddle-node equilibrium state

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). 

Fig. 11.2.4. Planar bifurcation of a saddle-node equilibrium state with 2 > 0.

GLOBAL BIFURCATIONS AT THE DISAPPEARANCE OF SADDLE-NODE EQUILIBRIUM STATES AND PERIODIC ORBITS... 

To study such bifurcations one should understand the structure of the limit set into which the periodic orbit transforms when the stability boundary is approached. In particular, such a limit set may be a homoclinic loop to a saddle or to a saddle-node equilibrium state. In another bifurcation scenario (called the blue sky catastrophe ) the periodic orbit approaches a set composed of homoclinic orbits to a saddle-node periodic orbit. In this chapter we consider homoclinic bifurcations associated with the disappearance of the saddle-node equilibrium states and periodic orbits. Note that we do not restrict our attention to the problem on the stability boundaries of periodic orbits but consider also the creation of invariant two-dimensional tori and Klein bottles and discuss briefly their routes to chaos. 

Bifurcations of a homoclinic loop to a saddle-node equilibrium state... 

Fig. 12.2.2. A nontransverse tangency of the unstable and strong-stable manifolds of a saddle-node fixed point may be obtained by a small time-periodic perturbation of the system with an on-edge homoclinic loop to a saddle-node equilibrium state, as shown in Fig. 12.1.4.

A periodic orbit merges with a homoclinic loop r e) to a saddle-node equilibrium state Og, where r(e) W (Oe). 

In this case the limit of a periodic trajectory as e —0 is a homoclinic cycle r composed of a simple saddle-node equilibrium state and its... 

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. 

It follows from the above theorem that a rough system on the plane may possess only rough equilibrium states (nodes, foci and saddles) and rough limit cycles. As for separatrices of saddles, they either tend asymptotically to a node, a focus, or a limit cycle in forward or backward time, or leave the region G after a finite interval of time. 

Fig. 8.1.3. A nontransverse homociinic loop F to a saddle node. The separatrix enters the equilibrium state along its strongly stable manifold.

The phase portraits for systems of dimension two and higher are illustrated in Figs. 11.2.4-11.2.7, respectively. Here, when l2 < 0, there are two rough equilibrium states a stable node and a saddle that approach each other as i2M... 

Consider a one-parameter family of (r > 2) smooth dynamical systems in (n > 1). Suppose that when the parameter vanishes the system possesses a non-rough equilibrium state at the origin with one characteristic exponent equal to zero and the other n exponents lying to the left of the imaginary axis. We suppose also that the equilibrium state is a simple saddle-node, namely the first Lyapimov value I2 is not zero (see Sec. 11.2). Without loss of generality we assume /2 > 0. 

In terms of the original variable (p — — ut, the stationary value of (the equilibrium state of system (12.1.9)) corresponds to an oscillatory regime with the same frequency as that of the external force. The periodic oscillations of (the limit cycle in (12.1.9)) correspond to a two-frequency regime. Hence, the above bifurcation scenario of a limit cycle from a homoclinic loop to a saddle-node characterizes the corresponding route from synchronization to beat modulations in Eq. (12.1.7). 

Fig. 12.1.7. The boundary of the stability region of the periodic orbit born at the bifurcation of the on>edge homoclinic loop to a degenerate saddle-node corresponds to the homoclinic loop of the border saddle equilibrium state Oi.

The bifurcation diagram is presented in Fig. 12.4.4. When e > 0, the saddle-node O disappears, while the equilibrium state O is decomposed into... 

This is a one-dimensional system which may have stable and unstable equilibrium states corresponding to stable and saddle equilibrium states of the entire system (12.4.6) or (12.4.7). The evolution along Meq is either limited to one of the stable points, or it reaches a small neighborhood of the critical values of X. Recall, that we consider x as a governing parameter for the fast system and critical values of x are those ones which correspond to bifurcations of the fast system. In particular, at some x two equilibrium states (stable and saddle) of the fast system may coalesce into a saddle-node. This corresponds to a maximum (or a minimum) of x on Meq, so the value of x cannot further... 

The Cherry flow is a flow on a two-dimensional torus with two equilibrium states a saddle and an unstable node both unstable separatrices are stable one stable separatrix is a-limit to a node and the other lies in the closure of the unstable separatrices and it is P -stable [see Fig. 13.7.4(a)]. The closure of the unstable separatrices is a quasiminimal set which contains the saddle O and a continuum of unclosed P-stable trajectories. The rotation number for such flows is defined in the same way as for flows on a torus without equilibrium states. Since there is no periodic orbits in a Cherry flow. 

Fig. 13.7.20. The bifurcation diagram for the case where both equilibrium states of the heteroclinic cycle are saddles (see Fig. 13.7.12) provided Ai,2 > 0, i/i > 1, 1/2 < 1 and U1U2 > 1- The system has one limit cycle in regions 1-3, two limit cycles in region 4, none in regions 5-7. A pair of limit cycles are born from a saddle-node on the curve SN the unstable one becomes a homoclinic loop on the curve L2, whereas the stable limit cycle terminates on Li.

The second example exhibits a stable equilibrium state which merges with a saddle to spawn a saddle-node. Denote by F, the only imstable trajectory leaving the saddle-node as f -f 00, and its limit set by n(F). If Cl T) is an... 

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. 

Fig. C.2.6. The x-coordinate of the equilibrium state versus z in the fast planar system at / = 5 and e = 0. AH and SN denote, respectively, the Andronov-Hopf and the saddle-node bifurcations of the equilibria.

This approach is a rather general one. Its advantage is that when the rescaling procedure has been carried out, many resonant monomials disappear. The most trivial example is a saddle-node bifurcation with a single zero eigenvalue. In this case the center manifold is one-dimensional. The Taylor expansion of the system near the equilibrium state may be written in the following form... 

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