Saddle-node bifurcation for maps

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Yuri A. Kuznetsov (2008), Scholarpedia, 3(4):4399. doi:10.4249/scholarpedia.4399 revision #37507 [link to/cite this article]

Curator: Dr. Yuri A. Kuznetsov, Department of Mathematics, Utrecht University, The Netherlands

Saddle-node bifurcation for maps
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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.

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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.

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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.

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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.

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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.

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Other Cases

When the saddle-node bifurcation occurs for the Nth-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.

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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.
  • 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.


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External Links

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See Also

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


Yuri A. Kuznetsov (2008) Saddle-node bifurcation for maps. Scholarpedia, 3(4):4399, (go to the first approved version)
Created: 7 July 2007, reviewed: 6 December 2007, accepted: 10 April 2008
Invited by: Prof. James Meiss, Applied Mathematics University of Colorado
Action editor: Prof. James Meiss, Applied Mathematics University of Colorado
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