Neimark-Sacker bifurcation
Neimark-Sacker bifurcation is the birth of a closed invariant curve from a fixed point in dynamical systems with discrete time (iterated maps), when the fixed point changes stability via a pair of complex eigenvalues with unit modulus. The bifurcation can be supercritical or subcritical, resulting in a stable or unstable (within an invariant two-dimensional manifold) closed invariant curve, respectively. When it happens in the Poincare map of a limit cycle, the bifurcation generates an invariant two-dimensional torus in the corresponding ODE.
Definition
Consider a map \[ x \mapsto f(x,\alpha),\ \ \ x \in {\mathbb R}^n \] depending on a parameter \(\alpha \in {\mathbb R}\ ,\) where \(f\) is smooth.
- Suppose that for all sufficiently small \(|\alpha|\) the system has a family of fixed points \(x^0(\alpha)\ .\)
- Further assume that its Jacobian matrix \(A(\alpha)=f_x(x^0(\alpha),\alpha)\) has one pair of complex eigenvalues
\[ \lambda_{1,2}(\alpha)=r(\alpha) e^{\pm i\theta(\alpha)} \] on the unit circle when \(\alpha=0\ ,\) i.e., \(r(0)=1\) and \(0 < \theta(0) < \pi\ .\) Then, generically, as \(\alpha\) passes through \(\alpha=0\ ,\) the fixed point changes stability and a unique closed invariant curve bifurcates from it. This bifurcation is characterized by a single bifurcation condition \(|\lambda_{1,2}|=1\) (has codimension one) and appears generically in one-parameter families of smooth maps.
Two-dimensional Case
To describe the bifurcation analytically, consider the map above with \(n=2\ ,\) \[ \left(\begin{array}{c} x_1 \\ x_2 \end{array}\right) \mapsto \left(\begin{array}{c} f(x_1,x_2,\alpha) \\ f(x_1,x_2,\alpha) \end{array}\right) \ .\] If the following nondegeneracy conditions hold:
- (NS.1) \( e^{ik\theta_0} \neq 1\) for \(k=1,2,3\ ,\) and \(4\ ,\) where \( \theta_0=\theta(0)\) (no strong resonances);
- (NS.2) \(r'(0) \neq 0\ ,\)
then this map is locally conjugate near the fixed point to the normal form, that can be written using a complex coordinate \(z=y_1 + iy_2\) as \[ z \mapsto (1 + \beta)e^{i \theta(\beta)}z + c(\beta)z|z|^2 + O(|z|^4) \ ,\] where \(y=(y_1,y_2)^T \in {\mathbb R}^2\) and \( \beta \in {\mathbb R}\) is the new parameter; see, for example, Iooss (1979) or Arnold (1983).
If, moreover,
- (NS.3) \(d(0)={\rm Re}[e^{-i\theta_0} c(0)] \neq 0\ ,\) where \(d(0)\) is the first Lyapunov coefficient (see below), then
- If \(d(0) < 0\ ,\) the normal form has a fixed point at the origin, which is asymptotically stable for \(\beta \leq 0\) (weakly at \(\beta=0\)) and unstable for \(\beta>0\ .\) Moreover, there is a unique and stable closed invariant curve that exists for \(\beta>0\) and has radius \(O(\sqrt{\beta})\ .\) This is a supercritical Neimark-Sacker bifurcation (see Figure 1).
- If \(d(0) > 0\ ,\) the fixed point at the origin in the normal form is asymptotically stable for \(\beta<0\) and unstable for \(\beta \geq 0\) (weakly at \(\beta=0\)), while a unique and unstable closed invariant curve of radius \(O(\sqrt{-\beta})\) exists for \(\beta <0\ .\) This is a subcritical Neimark-Sacker bifurcation (see Figure 2).
Note that in some books the wrong nondegeneracy condition \({\rm Re}[c(0)] \neq 0\) is given instead of (NS.3). Applying this wrong condition, one can draw false conclusions about the direction of birth and stability of the closed invariant curve.
Arnold Tongues
The smoothness of the closed invariant curve for a fixed parameter value is only finite - even if the original map is infinitely-differentiable - but increases if \( \beta \to 0\ .\) The \(O(|z|^4)\)-terms in the normal form cannot be truncated, since they effect the orbit structure on the closed invariant curve. When these terms are present and depend generically on \( {\rm arg}(z)\ ,\) the orbits on the invariant curve can either be all everywhere dense or there exists only a finite number of periodic orbits. The orbit structure varies with \( \beta \ :\) Exactly two periodic orbits exist in some open parameter interval but disappear on its borders through the saddle-node bifurcation for maps. A generic map exhibits near the Neimark-Sacker bifurcation an infinite number of these bifurcations of cycles in the closed invariant curve, corresponding to the borders of the infinite number of such intervals.
Let \( \lambda=\lambda_1\) and consider \({\rm Re}\; \lambda \) and \({\rm Im}\; \lambda \) as new independent parameters. On the plane of these parameters, the unit circle \(|\lambda| = 1 \) corresponds to the Neimark-Sacker bifurcation. Staying away from the strong resonances and assuming that the bifurcation is supercritical, we get a stable closed invariant curve for parameters \(({\rm Re}\; \lambda, {\rm Im}\; \lambda)\) outside the unit circle. Parameter regions, in which pairs of periodic orbits on the closed invariant curve exist, approach the unit circle at all rational points \[ \lambda =e^{i\theta},\ \ \theta=\frac{2\pi p}{q}, \] as narrow tongues with the \(O((|\lambda|-1)^{(q-2)/2}\)-width. These Arnold tongues are bounded by the saddle-node bifurcation curves for the cycles. Far from the Neimark-Sacker bifurcation, different tongues can intersect. At such parameter values, the closed invariant curve does not exist and two independent saddle-node bifurcations happen with unrelated remote cycles.
Multi-dimensional Case
In the \(n\)-dimensional case with \(n \geq 3\ ,\) the Jacobian matrix \(A_0=A(0)\) generically has
- a simple pair of critical eigenvalues \(\lambda_{1,2}=e^{\pm i \theta_0}\) such that \( e^{ik\theta_0} \neq 1\) for \(k=1,2,3\ ,\) and \(4\ ,\) as well as
- \(n_s\) eigenvalues with \(|\lambda_j| < 1\ ,\) and
- \(n_u\) eigenvalues with \(|\lambda_j| > 1\ ,\) with \(n_s+n_u+2=n\ .\)
According to the Center Manifold Theorem for maps, there is a family of smooth two-dimensional invariant manifolds \(W^c_{\alpha}\) near the origin. The \(n\)-dimensional map restricted on \(W^c_{\alpha}\) is two-dimensional, hence has the normal form above and demonstrates the described bifurcation.
First Lyapunov Coefficient
Whether a Neimark-Sacker bifurcation is subcritical or supercritical is determined by the sign of the first Lyapunov coefficient \(d(0)\ .\) This coefficient can be computed at \(\alpha=0\) as follows. Assume that \( x^0(0)=0\) and write the Taylor expansion of \(f(x,0)\) at \(x=0\) as \[ f(x,0)=A_0x + \frac{1}{2}B(x,x) + \frac{1}{6}C(x,x,x) + O(\|x\|^4), \] where \(B(x,y)\) and \(C(x,y,z)\) are the multilinear functions with components \[ \ \ B_j(x,y) =\sum_{k,l=1}^n \left. \frac{\partial^2 f_j(\xi,0)}{\partial \xi_k \partial \xi_l}\right|_{\xi=0} x_k y_l \ ,\] \[ C_j(x,y,z) =\sum_{k,l,m=1}^n \left. \frac{\partial^3 f_j(\xi,0)}{\partial \xi_k \partial \xi_l \partial \xi_m}\right|_{\xi=0} x_k y_l z_m \ ,\] where \(j=1,2,\ldots,n\ .\) Let \(q\in {\mathbb C}^n\) be a complex eigenvector of \(A_0\) corresponding to the eigenvalue \(e^{i\theta_0}\ :\) \(A_0q=e^{i\theta_0} q\ .\) Introduce also the adjoint eigenvector \(p \in {\mathbb C}^n\ :\) \(A_0^T p = e^{- i\theta_0} p\ ,\) \( \langle p, q \rangle =1\ .\) Here \(\langle p, q \rangle = \bar{p}^Tq\) is the inner product in \({\mathbb C}^n\ .\) Then \[ c(0)= \frac{1}{2} \left[\langle p,C(q,q,\bar{q}) \rangle + 2 \langle p, B(q,(I_n - A_0)^{-1}B(q,\bar{q}))\rangle + \langle p, B(\bar{q},(e^{2i\theta_0}I_n-A_0)^{-1}B(q,q))\rangle \right], \] where \(I_n\) is the unit \(n \times n\) matrix, and \[ d(0)={\rm Re}[e^{-i\theta_0} c(0)] \ .\] A formula equivalent to the one above was first derived by Iooss et al.(1981) using asymptotic expansions for the branching closed invariant curve, while a derivation based on the center manifold reduction can be found in Kuznetsov (2004). Note that the value (but not the sign) of \(d(0)\) depends on the scaling of the eigenvector \(q\ .\) The normalization \( \langle q, q \rangle =1\) is one of the options to remove this ambiguity. Standard bifurcation software (e.g. MATCONT) computes \(d(0)\) automatically.
Torus Bifurcation
Suppose that the Neimark-Sacker bifurcation occurs in the Poincare map of a limit cycle in ODE, so that the fixed point corresponding to the limit cycle has a pair of simple eigenvalues \( \mu_{1,2}=e^{\pm i \theta_0} \) and all formulated above genericity conditions hold. Then a unique two-dimensional invariant torus bifurcates from the cycle, while it changes stability. The intersection of the torus with the Poincare section corresponds to the closed invariant curve. The torus bifurcation is sometimes called the secondary Hopf bifurcation.
The torus bifurcation can occur near the fold-Hopf bifurcation and is always present near the Hopf-Hopf bifurcation of equilibria in ODEs.
Strong Resonances
The torus bifurcation of limit cycles was known to Andronov. The first paper on this bifurcation by Neimark (1959) contained an error: He underestimated the role of the strong resonances and omitted by mistake the corresponding nondegeneracy conditions. However, these conditions are not merely technical, since near such resonances more than one closed invariant curve can appear, or no such curves may exist at all. The first complete proof in the absence of all strong resonances was given by Sacker (1964), who discovered the bifurcation independently. For the modern theory of strong resonances as the two-parameter phenomena, see Arnold (1983).
References
- Ju.I. Neimark (1959) On some cases of periodic motions depending on parameters. Dokl. Akad. Nauk SSSR 129, 736-739 [in Russian].
- R. Sacker (1964) On invariant surfaces and bifurcation of periodic solutions of ordinary differential equations. Report IMM-NYU 333, New York University.
- G. Iooss (1979) Bifurcations of Maps and Applications, North Holland, Amsterdam.
- G. Iooss, A. Arneodo, P. Coullet, and C. Tresser. Simple computation of bifurcating invariant circles for maps. In: D. Rand and L.-S. Young (eds.) "Dynamical Systems and Turbulence", Lecture Notes in Mathematics 898 (1981), 192-211.
- V.I. Arnold (1983) Geometrical Methods in the Theory of Ordinary Differential Equations. Grundlehren Math. Wiss., 250, Springer.
- L.P. Shilnikov, A.L. Shilnikov, D.V. Turaev, and L.O. Chua (2001) Methods of Qualitative Theory in Nonlinear Dynamics. Part II, World Scientific.
- Yu.A. Kuznetsov (2004) Elements of Applied Bifurcation Theory, Springer, 3rd edition.
Internal references
- John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.
- Andrey Shilnikov and Dmitry Turaev (2007) Blue-sky catastrophe. Scholarpedia, 2(8):1889.
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- John Guckenheimer and Yuri A. Kuznetsov (2007) Fold-Hopf bifurcation. Scholarpedia, 2(10):1855.
- Willy Govaerts, Yuri A. Kuznetsov, Bart Sautois (2006) MATCONT. Scholarpedia, 1(9):1375.
- Yuri A. Kuznetsov (2008) Saddle-node bifurcation for maps. Scholarpedia, 3(4):4399.
- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
External Links
See Also
Bifurcations, Center Manifold Theorem, Dynamical Systems, Fixed points, MATCONT,