Curator and Contributors

1.00 - Yuri A. Kuznetsov

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.

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.

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

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