Saddle-node bifurcation for maps

• Prof. Yuri A. Kuznetsov, Department of Mathematics, Utrecht University, The Netherlands

Figure 1: Saddle-node bifurcation for the map \(x \mapsto g(x,\alpha)=x^2+x-\alpha\ .\)

A saddle-node bifurcation in dynamical systems with discrete time (iterated maps) is a birth of two fixed points of the generating map. This occurs when the critical fixed point has one eigenvalue \(+1\ .\) This phenomenon is also called fold or limit point bifurcation of maps. This bifurcation is a discrete version of the saddle-node bifurcation in flows (ODEs), where an equilibrium has one zero eigenvalue.

Definition

Consider a discrete-time dynamical system generated by 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 at \(\alpha=0\) the system has a fixed point \(x^0=0\ .\)
• Further assume that its Jacobian matrix \(A_0=f_x(0,0)\) has a simple eigenvalue \(\mu_{1}=1 \ .\)

Then, generically, as \(\alpha\) passes through \(\alpha=0\ ,\) two fixed point are born form a critical fixed point (see Figure 1). This bifurcation is characterized by a single bifurcation condition \(\mu_1=1\) (has codimension one) and appears generically in one-parameter families of smooth maps. The critical fixed point \( x^0 \) is a multiple (double) root of the equation \( f(x,0)=x \ .\)

One-dimensional Case

To describe the bifurcation analytically, consider the map above with \(n=1\ ,\) \[ x \mapsto f(x,\alpha), \ \ \ x \in {\mathbb R} \ .\] The bifurcation condition in this case is \(f_x(0,0)=1\ .\) If the following nondegeneracy conditions hold:

• (SN.1) \(a(0)=\frac{1}{2}f_{xx}(0,0) \neq 0\ ,\)
• (SN.2) \(f_{\alpha}(0,0) \neq 0\ ,\)

then this one-parameter family of maps (with parameter \(\alpha\)) is locally conjugate near the origin to the following one-parameter family (with parameter \(\beta\)): \[ y \mapsto \beta + y + \sigma y^2 \ ,\] where \(y \in {\mathbb R},\ \beta \in {\mathbb R}\ ,\) and \(\sigma= {\rm sign}\ a(0) = \pm 1\ .\) This latter family of maps is called the normal form for the saddle-node bifurcation (although it is not a normal form in the sense of the article with that title).

The normal form has no fixed points for \(\sigma \beta > 0\) and two fixed points (one stable and one unstable) \(y^{1,2}=\pm \sqrt{-\sigma \beta}\) for \(\sigma \beta<0\ .\) At \(\beta=0\ ,\) there is one critical fixed point \(y^0=0\) with eigenvalue 1.

Multidimensional Case

In the \(n\)-dimensional case with \(n \geq 2\ ,\) the Jacobian matrix \(A_0\) at the saddle-node bifurcation has

• a simple eigenvalue \(\mu_{1}=1\ ,\) as well as
• \(n_s\) eigenvalues with \(|\mu_j| < 1\ ,\) and
• \(n_u\) eigenvalues with \(|\mu_j| > 1\ ,\)

with \(n_s+n_u+1=n\ .\) According to the Center Manifold Theorem, there is a family of smooth one-dimensional invariant manifolds \(W^c_{\alpha}\) near the origin. The \(n\)-dimensional system restricted on \(W^c_{\alpha}\) is one-dimensional, hence has the normal form above.

Quadratic Coefficient

The quadratic coefficient \(a(0)\ ,\) which is involved in the nondegeneracy condition (SN.1), can be computed for \(n \geq 1\) as follows. Write the Taylor expansion of \(f(x,0)\) at the fixed point \(x=0\) as \[ f(x,0)=A_0x + \frac{1}{2}B(x,x) + O(\|x\|^3) \ ,\] where \(B(x,y)\) is the bilinear function with components \[ \ \ B_j(x,y) =\sum_{k,l=1}^n \left. \frac{\partial^2 f_j(\xi,0)}{\partial \xi_k \partial \xi_l}\right|_{\xi=0} x_k y_l \ ,\] where \(j=1,2,\ldots,n\ .\) Let \(q\in {\mathbb R}^n\) be a critical eigenvector of \(A_0\ :\) \(A_0q=q, \ \langle q, q \rangle =1\ ,\) where \(\langle p, q \rangle = p^Tq\) is the standard inner product in \({\mathbb R}^n\ .\) Introduce also the adjoint eigenvector \(p \in {\mathbb R}^n\ :\) \(A_0^T p = p, \ \langle p, q \rangle =1\ .\) Then \[ a(0)= \frac{1}{2} \langle p, B(q,q))\rangle = \left.\frac{1}{2} \frac{d^2}{d\tau^2} \langle p, f(\tau q,0) \rangle \right|_{\tau=0} \ .\] Standard bifurcation software (e.g. MATCONT) computes \(a(0)\) automatically.

Other Cases

When the saddle-node bifurcation occurs for the \(N\)th-iterate of a map, two periodic orbits (or cycles) of period \(N\) are generically born. If a saddle-node bifurcation occurs for a Poincare map (or its iterate) defined by an ODE, it implies the birth of two limit cycles.

References

• V.I. Arnold (1983) Geometrical Methods in the Theory of Ordinary Differential Equations. Grundlehren Math. Wiss., 250, Springer
• D. Whitley (1983) Discrete dynamical systems in dimension one and two. Bull. London Math. Soc. 15, 177-217.
• J. Guckenheimer and P. Holmes (1983) Nonlinear Oscillations, Dynamical systems and Bifurcations of Vector Fields. Springer
• Yu.A. Kuznetsov (2004) Elements of Applied Bifurcation Theory, Springer, 3rd edition.
• S. Newhouse, J. Palis and F. Takens (1983) Bifurcations and stability of families of diffeomorphisms. Inst. Hautes Études Sci. Publ. Math. 57, 5-71.

Internal references

• John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.

• Yuri A. Kuznetsov (2007) Conjugate maps. Scholarpedia, 2(12):5420.

• James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.

• Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.

• Willy Govaerts, Yuri A. Kuznetsov, Bart Sautois (2006) MATCONT. Scholarpedia, 1(9):1375.

• James Murdock (2006) Normal forms. Scholarpedia, 1(10):1902.

• Yuri A. Kuznetsov (2006) Saddle-node bifurcation. Scholarpedia, 1(10):1859.

• Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

See Also

Saddle-node bifurcation in flows, Bifurcations, Center manifold theorem, Dynamical systems, Fixed points, MATCONT,

Sponsored by: Prof. James Meiss, Applied Mathematics University of Colorado, Boulder, CO, USA

Reviewed by: Anonymous

Accepted on: 2008-02-11 23:16:23 GMT