PlanetMath: cycle

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cycle

(Definition)

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.

"cycle" is owned by Daume.

(view preamble)

Other names:

periodic solution, stable periodic solution, unstable periodic solution, asymptotically stable periodic solution

Also defines:

period, stable cycle, unstable cycle, asymptotically stable cycle

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Cross-references: neighborhood, metric, ODE, flow, positive, equilibrium point, solution, vector field, ordinary differential equation, autonomous
There are 25 references to this entry.

This is version 3 of cycle, born on 2005-02-06, modified 2007-01-23.
Object id is 6721, canonical name is Cycle4.
Accessed 6785 times total.

Classification:

AMS MSC:	34C07 (Ordinary differential equations :: Qualitative theory :: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramif)
	34A12 (Ordinary differential equations :: General theory :: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions)

Pending Errata and Addenda

None.[ View all 1 ]

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