文档介绍：EMAT33100 - Nonlinear Dynamics & Chaos 1
EMAT33100: Nonlinear Dynamics & Chaos
Lecture Notes
by
Hinke Osinga (.******@bristol)
1 What is a dynamical system?
A dynamical system is a system whose behaviour can be described by an evolution operator
Φt : X → X
defined on a space X for all t ∈ T .ThespaceX is called the state space or phase space.
The space T , or time, can be R, in which case we speak of a continuous time system, or
T = Z, which corresponds to a discrete time system. The evolution operator must be such
that the following two conditions hold for any initial condition x0 ∈ X and any t, s ∈ T :
0
1. Φ(x0)=x0 (“no time, no evolution”),
t+s t s
2. Φ(x0)=Φ(Φ(x0)) (“determinism”).
If T = R, then the dynamical system is a continuous time system, which is typically given
as a vector field
x˙= f(x), (1)
where f is a function defined on X. The evolution operator describes the flow of the vector
field. Special solutions of a vector field are:
• Equilibria or rest points: Points x∗∈ X with f(x∗)=0.
• Periodic orbits: For a point x∗∈ X there exists τ∈ R with τ>0 such that
Φτ(x∗)=x∗. The periodic orbit is defined as the closed curve
{Φt(x∗) | 0 ≤ t<τ∗},
where τ∗> 0 is the smallest number τ such that Φτ(x∗)=x∗, and it is called the
period of the periodic orbit.
2 Hinke Osinga
If T = Z, then the dynamical system is a discrete system and defined by a map
x → g(x), (2)
where g is a function defined on X. The evolution operator is the map g itself and Φt(x)=
Φn(x)=gn(x). Special solutions of a discrete system are:
• Equilibria or fixed points: Points x∗∈ X with g(x∗)=x∗.
• Periodic orbits: For a point x∗∈ X there exists n ∈ Z with n>0 such that
gn(x∗)=x∗, while gi(x∗) = x∗ for all 0 <i<n. The periodic orbit is defined as the
set of n points
{gi(x∗) | 0 ≤ i<n},
and n is called the period of the periodic orbit.
Given the evolution operator, we can find out what happens to x0 as time evolves. With
time increasing, one wou