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