Shilnikov saddle-node bifurcation

• Leonid Pavlovich Shilnikov, Institute for Applied Mathematics and Cybernetics, Nizhni Novgorod, Russia

• Andrey Shilnikov, Dept Mathematics, GSU, Atlanta, GA

Figure 1: Illustration to Theorem 3: the saddle-saddle is formed by two saddles of topological types (2,1) and (1,2) in \(R^3\) respectively. Its stable, \( W^s \), and and unstable, \( W^s \), manifolds cross transversally along two homoclinic orbits. As soon as the equilibrium is gone, there emerges a new-born hyperbolic set with dynamics conjugate to a suspension over a Bernoulli scheme on two symbols.

Saddle-saddle or Shilnikov saddle-node bifurcation gives rise to complex dynamics in a system after a merger of two saddles connected globally by \( m \) (\( \ge 2 \)) heteroclinic orbits. The later ones become the transverse homoclinic connections of the saddle-saddle at the bifurcation, as illustrated in Fig. <ref>fig1</ref>. After the saddle-saddle vanishes through this codimension-one bifurcation, the system's dynamics becomes conjugate to a suspension over a Bernoulli scheme on same \( m \) symbols.

Single zero exponent

Let us consider an \( n\)-dimensional, sufficiently smooth system that has a structurally unstable equilibrium state \(O\) at the origin with eigenvalues \( Re \lambda_i \neq 0\) ( \( i=1,...,n-1\)), and \( Re \lambda_n=0\). Near the origin, the system can then be written in the form [L. Shilnikov et al., 1998 and 2001]:

<math e1>

\dot x=f(x,y), \qquad \dot y = Ay+g(x,y), </math> where \(f,\,g\) vanish at the origin along with their first partials. It is supposed that in the Taylor expansion \[f(x,\varphi(x))=l_2 x^2 + l_3 x^3 \cdots \], the first Lyapunov coefficient \(l_2 \neq 0\); here \(y=\varphi(x)\) is the solution of the equation \(Ay+g(x,y)=0\); here \(A\) is an \((n-1)\times (n-1)\) matrix.

The behavior of the solutions of the system (1) near the origin is similar to that of the following truncated normal form

<math e2> \dot x=\mu+x^2, \qquad \dot y = Ay.

</math> Depending on the spectrum of the matrix \(A\), the following cases are possible:

1) The case \(Re\,\lambda_i<0\), \(i=1,...,n-1\), is illustrated in Fig. <ref>fig2</ref>.

Figure 2: Local saddle-node bifurcation in \(R^3\): a stable node merges with a saddle of topological type (2,1)

Figure 3: Homoclinic saddle-node bifurcation generates a stable periodic orbit.

Here, the \((n-1)\)-dimensional non-leading or strongly stable manifold \(W^{ss}\) breaks a neighborhood of the equilibrium state into the two regions: stable (or node) and saddle. The saddle one contains the trajectories \(\Gamma^u\) that originate from \(O\), or converge to it as \( t \to -\infty\), while in the stable node region the trajectories converge to the origin in the forward time\[ t \to +\infty\] . Under small, smooth perturbations of the vector field, \(O\) either vanishes or decouples into two equilibrium states: a stable node and a saddle. This is the local saddle-node bifurcation.

2) The case \(Re\,\lambda_i>0\), \(i=1,...,n-1\), is reduced to the previous one by the time change \(t \to -t\), which makes the unstable node stable, while preserving the saddle structure of the other equilibrium state.

3) The case \(Re\,\lambda_i>0\), \( i=1,...,k\), and \(Re\,\lambda_j>0\), \(j=k+1,...,n-1\). The corresponding equilibrium state is called a saddle-saddle, as it is the result of a merger of two saddles of the proper topological types, see Figs. <ref>fig1</ref> and <ref>fig5</ref>. It has a \((k+1)\)-dimensional stable manifold \(W^s\), which is diffeomorphic to a half-space \( \{\hat R^{k}\, | (x,y_1,...,y_k), x<0\} \), as well as an unstable one \(W^u\) diffeomorphic to a half-space \( \{\hat R^{n-k}\, | (x,y_{1},...,y_{n-k-1}), x>0\} \).

Figure 4: Single transverse homoclinic connection of a saddle-saddle at the intersection of it stable and unstable manifolds becomes a saddle periodic orbit.

Saddle-node bifurcation for synchronization

Back in the early 30's of the last century, A. Andronov and A. Vitt studied the phenomenon of the 1:1 resonance in the periodically forced Van der Pol equation \[ \ddot x - \mu(1 - x^2) \dot x + \omega^2_0x = A \sin \omega t, \] where \(\mu \ll 1\) and \(|\omega_0 -\omega| \sim \mu\). They had discovered that the disappearance of the stable equilibrium state is accompanied by the emergence of a stable limit cycle in the system: or in other terms, they discovered the transition mechanism between synchronization (phase locking) and modulation (beating oscillations). A rigorous mathematical explanation of this phenomenon was later given by A. Andronov and E. Leontovich with the use of the tools of global bifurcations.

Theorem 1. Let a two-dimensional system have an equilibrium state \( O \) of the saddle-node type with the characteristic exponents \( \lambda_1 <0\) and \( \lambda_2 =0\) such that its unstable separatrix \( \Gamma^u \) comes back to \( O \) as \( t \to +\infty \) not in \( W^{ss}.\) Then, as the saddle-node has vanished, a single, stable periodic orbit emerges from its homoclinic loop \( \bar \Gamma \).

The generalization of this bifurcation for a high-dimensional case was done by L. Shilnikov [1963]. Its stages are sketched in Fig. <ref>fig3</ref>.

Saddle-saddles

In addition to the saddle-node L. Shilnikov proposed and examined another peculiar homoclinic bifurcation of a saddle-saddle in [1969]:

Theorem 2. Let \( \Gamma \in W^u \cap W^u\) and not in \( \partial W^s \cap \partial W^s \), i.e., \( W^u \) and \( W^s \) cross along \( \Gamma \) transversally. Then, after the saddle-saddle has vanished, \( \bar \Gamma \) becomes a single, saddle periodic orbit (<ref>fig4</ref>).

The proof of this theorem called for a creation of the special technique of solving the boundary value problem known, today as the Shilnikov coordinates, which is especially helpful for proving the existence of complex hyperbolic dynamics generated by saddle orbits.

Theorem 3. Let \( W^u \) and \( W^s \) of a saddle-saddle cross transversally along \( \Gamma_1, \cdots, \Gamma_m \) (see Fig. <ref>fig1</ref> ). Then, after the saddle-saddle has disappeared, the hyperbolic set conjugate to a suspension over a subshift on \( m \) symbols is born.

Figure 5: Leading eigenvalues of the equilibrium involved in the bifurcation.

In general, in a codimension-one case, \( W^u \) and \( W^s \) of a saddle-saddle may intersect transversally over a countable set of such homoclinic orbits; i.e. the closure of this set includes countably many saddle periodic orbits. If so, the disappearance of the saddle-saddle gives rise to the emergence of a one-dimensional hyperbolic set, to describe which one needs to employ topological Markov chains with finite numbers of states. [Afraimovich and Shilnikov, 1983]

Applications and Examples

The complex set resulting from the Shilnikov saddle-saddle bifurcation set is not an attractor, in general. That is perhaps the reason why a realistic application of this simple co-dimension-1 bifurcation is yet wanted. An attempt was undertaken by Glendenning and Sparrow [1996] whose underlying idea was to get rid of the saddle point at the origin in a Lorenz attractor-alike system through the considered bifurcation, right when the saddle-saddle possesses a homoclinic butterfly formed simultaneously by two homoclinic loops (see Fig. <ref>fig6</ref> ). This would have made the strange attractor of the system truly hyperbolic [Afraimovich et al, [1977, 1983]], if its orbits did not get away from it down along the z-axes. It follows from the later references that since the pre-images of the stable manifold of the pre-existing saddle are dense everywhere on the attractor, then the phase point will inevitably come by arbitrarily close to the z-axis sooner or later even when the saddle is gone. The other complication is that when the point does fall down far enough, it will get re-injected back (Fig. <ref>fig7</ref>) so that while lowering to the strange attractor from above it will be turning around the z-axes too. This will create the distinctive hooks [Shilnikov, [1993]] in the shape of the 2D Poincare mapping that indicate clearly that the Lorenz attractor no longer persists the necessary property of tranversality for the stable and unstable foliations.

Figure 6: Surrogate saddle-saddle with a pair of the homoclinic orbits in the modified Morioka-Simizu model [Shilnikov, [1993].

Figure 7: A Lorenz-like transient chaos in the modified Morioka-Simizu model where the saddle-saddle bifurcation has eliminated the equilibrium state at the origin.

References

• Afraimovich V.S. and Shilnikov L.P. [1983], Strange attractors and quasiattractors in Nonlinear Dynamics and Turbulence eds. by G.I. Barenblatt, G. Iooss and D.D. Joseph, Pitman, NY, 1-28.

• Afraimovich V.S., Bykov V.V. and Shilnikov L.P. [1977] On the appearance and structure of Lorenz attractor, DAN SSSR, 234, 336-339.

• Afraimovich V.S., Bykov V.V. and Shilnikov L.P. [1983] On the structurally unstable attracting limit sets of Lorenz attractor type. Tran. Moscow Math. Soc., 2, 153-215.

• Andronov A. A. and Vitt A. A. [1930] Zur Theorie des Mitmehmens von van der Pol. Archiv fur Elektrotechnik, Bd. XXIV, 99.

• Andronov A. A. and Leontovich E. A. [1937] Some cases of the dependence of the limit cycles upon parameters. Uchenye zapiski Gorkovskogo Universiteta 6, 3-24.

• Champneys A.R., Härterich J. and Sandstede B. A non-transverse homoclinic orbit to a saddle-node equilibrium. Ergodic Theory Dynam. Systems 16 (1996), no. 3, 431-450.

• Glendinning P. and Sparrow C. [1996] Shilnikov's saddle-node bifurcation. Int. J. Bifurcations & Chaos 6, 1153.

• Shilnikov L. P. [1963] Some cases of generation of periodic motion from singular trajectories. Math. USSR Sbornik 61(103), 443-466.

• Shilnikov L.P. [1969] On a new bifurcation of multidimensional dynamical systems. Sov. Math. Dokl. 10, 1389-1371.

• Shilnikov L.P., Shilnikov A., Turaev D. and Chua L. [1998] Methods of Qualitative Theory in Nonlinear Dynamics. Part I. World Scientific.

• Shilnikov L.P. Shilnikov A. Turaev D. and Chua L. [2001] Methods of Qualitative Theory in Nonlinear Dynamics. Part II.World Scientific.

• Shilnikov A.L. [1993] On bifurcations of the Lorenz attractor in the Shimizu-Morioka model. Physica D 62 (1-4), 338-346.

See Also

Bifurcations, Shilnikov bifurcation, Homoclinic Bifurcations, Homoclinic Orbits, Lorenz attractor, Smale Horseshoe, Saddle-focus, Saddle-node, Chaos

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Accepted on: 2008-04-23 04:22:19 GMT