cycle
Let
$$\dot{x}=f(x)$$ |
be an autonomous^{} ordinary differential equation^{} defined by the vector field $f:V\to V$ 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 $T$ such that $x(t)=x(t+T)$.
Let ${\varphi}_{t}(x)$ be the flow defined by the above ODE and $d$ be the metric of $V$ then:
A cycle, $\mathrm{\Gamma}$, is a stable cycle if for all $\u03f5>0$ there exists a neighborhood $U$ of $\mathrm{\Gamma}$ such that for all $x\in U$, $$.
A cycle, $\mathrm{\Gamma}$, is unstable cycle if it is not a stable cycle.
A cycle, $\mathrm{\Gamma}$, is asymptotically stable cycle if for all $x\in U$ where $U$ is a neighborhood of $\mathrm{\Gamma}$, ${lim}_{t\to \mathrm{\infty}}d({\varphi}_{t}(x),\mathrm{\Gamma})=0$.[PL]
example:
Let
$\dot{x}$ | $=$ | $-y$ | ||
$\dot{y}$ | $=$ | $x$ |
then the above autonomous ordinary differential equations with initial value condition $(x(0),y(0))=(1,0)$ has a solution which is a stable cycle. Namely the solution defined by
$x(t)$ | $=$ | $\mathrm{cos}t$ | ||
$y(t)$ | $=$ | $\mathrm{sin}t$ |
which has a period of $2\pi $.
References
- PL Perko, Lawrence: Differential Equations and Dynamical Systems^{} (Third Edition). Springer, New York, 2001.
Title | cycle |
Canonical name | Cycle12 |
Date of creation | 2013-03-22 15:00:51 |
Last modified on | 2013-03-22 15:00:51 |
Owner | Daume (40) |
Last modified by | Daume (40) |
Numerical id | 6 |
Author | Daume (40) |
Entry type | Definition |
Classification | msc 34A12 |
Classification | msc 34C07 |
Synonym | periodic solution |
Synonym | stable periodic solution |
Synonym | unstable periodic solution |
Synonym | asymptotically stable periodic solution |
Defines | period |
Defines | stable cycle |
Defines | unstable cycle |
Defines | asymptotically stable cycle |