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Equation Neighborhood Metric Space Positive Differential Equation Vector Field Flow Autonomous System Hyperbolic Equilibrium Point

1	Metric Space
1	Neighborhood
1	Positive
1	Equation
2	Differential Equation
2	Vector Field
4	Hyperbolic Equilibrium Point
5	Flow
6	Autonomous System

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Cycle

Let

$\displaystyle \dot{x}=f(x)$

be an autonomous ordinary differential equation defined by the vector field $f\colon V \to V$ then $x(t)\in V$ a solution of the system is a cycle(or periodic solution) if it is a closed solution which is not an equilibrium point. The period of a cycle is the smallest positive

such that

.
Let $\phi_t(x)$ be the flow defined by the above ODE and

be the metric of

then:
A cycle, $\Gamma$ , is a stable cycle if for all $\epsilon>0$ there exists a neighborhood

of $\Gamma$ such that for all $x\in U$ , $d(\phi_t(x),\Gamma)< \epsilon$ .
A cycle, $\Gamma$ , is unstable cycle if it is not a stable cycle.
A cycle, $\Gamma$ , is asymptotically stable cycle if for all $x\in U$ where

is a neighborhood of $\Gamma$ , $\lim_{t\to\infty}d(\phi_t(x),\Gamma)=0$ .[PL]

example:
Let

$\displaystyle \dot{x}$	$\displaystyle =$	$\displaystyle -y$
$\displaystyle \dot{y}$	$\displaystyle =$	$\displaystyle x$

then the above autonomous ordinary differential equations with initial value condition

has a solution which is a stable cycle. Namely the solution defined by

$\displaystyle x(t)$	$\displaystyle =$	$\displaystyle \cos t$
$\displaystyle y(t)$	$\displaystyle =$	$\displaystyle \sin t$

which has a period of $2\pi$ .

References

PL: Perko, Lawrence: Differential Equations and Dynamical Systems (Third Edition). Springer, New York, 2001.