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:<math>\frac{ dy }{ dt } = -x-x^3-\alpha y,~ \alpha \ne 0</math> :<math>\frac{ dy }{ dt } = -x-x^3-\alpha y,~ \alpha \ne 0</math>


<math>(0,0)</math> is the only equilibrium point. The lineariation at the equilibrium is <math>(0,0)</math> is the only equilibrium point. The linearization at the equilibrium is


:<math>J(0,0) = \begin{pmatrix} :<math>J(0,0) = \begin{pmatrix}

Revision as of 14:59, 14 May 2008

In mathematics, especially in the study of dynamical system, a hyperbolic equilibrium point or hyperbolic fixed point is a special type of fixed point.

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

Definition

Let

F : R n → R n {\displaystyle F:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}

be a C (that is, differentiable) vector field with fixed 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.

Example

Consider the nonlinear system

d x d t = y , {\displaystyle {\frac {dx}{dt}}=y,}
d y d t = − x − x 3 − α y ,   α ≠ 0 {\displaystyle {\frac {dy}{dt}}=-x-x^{3}-\alpha y,~\alpha \neq 0}

( 0 , 0 ) {\displaystyle (0,0)} is the only equilibrium point. The linearization at the equilibrium is

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

The eigenvalues of this matrix are − α ± α 2 − 4 2 {\displaystyle {\frac {-\alpha \pm {\sqrt {\alpha ^{2}-4}}}{2}}} . For all values of α ≠ 0 {\displaystyle \alpha \neq 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 ) {\displaystyle (0,0)} . When α = 0 {\displaystyle \alpha =0} , the system has a nonhyperbolic equilibrium at ( 0 , 0 ) {\displaystyle (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.


See also

References

  1. Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin/Cummings Publishing, Reading Mass. ISBN 0-8053-0102-X
  2. Equilibrium (Scholarpedia)


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