Hyperbolic equilibrium point

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In the study of dynamical systems, a hyperbolic equilibrium point or hyperbolic fixed point is a fixed point that does not have any center manifolds. Near a hyperbolic point the orbits of a two-dimensional, non-dissipative system resemble hyperbolas. This fails to hold in general. Strogatz^[1] notes that "hyperbolic is an unfortunate name – it sounds like it should mean 'saddle point' – but it has become standard." Several properties hold about a neighborhood of a hyperbolic point, notably^[2]

A stable manifold and an unstable manifold exist,
Shadowing occurs,
The dynamics on the invariant set can be represented via symbolic dynamics,
A natural measure can be defined,
The system is structurally stable.

Orbits near a two-dimensional saddle point, an example of a hyperbolic equilibrium.

Maps

If T : Rⁿ → Rⁿ is a C¹ map and p is a fixed point then p is said to be a hyperbolic fixed point when the differential DT(p) has no eigenvalues on the unit circle.

One example of a map that its only fixed point is hyperbolic is the Arnold Map or cat map:

${\begin{bmatrix}x_{{n+1}}\\y_{{n+1}}\end{bmatrix}}={\begin{bmatrix}1&1\\1&2\end{bmatrix}}{\begin{bmatrix}x_{n}\\y_{n}\end{bmatrix}}\quad {\text{modulo }}1$

Since the eigenvalues are given by

$\lambda _{{1}}={\frac {3+{\sqrt {5}}}{2}}>1$

$\lambda _{{2}}={\frac {3-{\sqrt {5}}}{2}}<1$

Flows

Let F : Rⁿ → Rⁿ be a C¹ (that is, continuously differentiable) vector field with a critical point p and let J denote the Jacobian matrix of F at p. If the matrix J has no eigenvalues with zero real parts then p is called hyperbolic. Hyperbolic fixed points may also be called hyperbolic critical points or elementary critical points.^[3]

The Hartman-Grobman theorem states that the orbit structure of a dynamical system in a neighbourhood of a hyperbolic equilibrium point is topologically equivalent to the orbit structure of the linearized dynamical system.

Example

Consider the nonlinear system

${\frac {dx}{dt}}=y,$

${\frac {dy}{dt}}=-x-x^{3}-\alpha y,~\alpha \neq 0$

(0, 0) is the only equilibrium point. The linearization at the equilibrium is

$J(0,0)={\begin{pmatrix}0&1\\-1&-\alpha \end{pmatrix}}$ .

The eigenvalues of this matrix are ${\frac {-\alpha \pm {\sqrt {\alpha ^{2}-4}}}{2}}$ . For all values of α ≠ 0, the eigenvalues have non-zero real part. Thus, this equilibrium point is a hyperbolic equilbrium point. The linearized system will behave similar to the non-linear system near (0, 0). When α = 0, the system has a nonhyperbolic equilibrium at (0, 0).

Comments

In the case of an infinite dimensional system - for example systems involving a time delay - the notion of the "hyperbolic part of the spectrum" refers to the above property.

Notes

↑ Strogatz, Steven (2001). Nonlinear Dynamics and Chaos. Westview Press.
↑ Ott, Edward (1994). Chaos in Dynamical Systems. Cambridge University Press.
↑ Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin/Cummings Publishing, Reading Mass. ISBN 0-8053-0102-X

References

Equilibrium at Scholarpedia, curated by Eugene M. Izhikevich.

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