The dimension of phase space and its implications

Consider the following dynamical system known as Duffing's oscillator

$$\dot{x} = y$$

$$\dot{y} = x-x^3- \delta y + \gamma \cos \omega t$$ (8)

It has the general form

$$\dot{X}=f(X,t)$$ (9)

which differs from (2) in that the function $f(\cdot)$ depends explicitly on time. Systems like these (2) are called autonomous, whereas systems like (10) are nonautonomous. The fixed points of a dynamical system are those vectors $X$ for which (2) or (10) evaluate to zero. In the case of a discrete mapping (7), $X$ is a fixed point if $X = f(X)$.

These points are called hyperbolic if the linear approximation of the system near them is not characterized by eigenvalues with null real parts (continuos-time) or with unit magnitude (discrete-time). Otherwise they are non-hyperbolic.

Hyperbolic points can be sources, drains, or saddle points depending on if nearby solutions diverge from the fixed point, converge to it, or diverge from it in certain neighborhoods whereas in others they converge to it. Non-hyperbolic points, characterized by eigenvalues with null real part or unitary magnitude, are called centers.

Two dimensional autonomous continuos-time dynamical systems can only have solutions that are fixed points or closed curves (conservative systems are restricted even more, since they cannot have sources nor drains) and this limits the range of behaviors they may exhibit. Either they are attracted to stable periodic solutions or repelled from them but, in essence, nothing more happens. Consequently, a continuos-time autonomous system requires more than two dimensions to exhibit chaos. This restriction does not hold for discrete maps or for nonautonomous systems. Specifically, chaos is possible for one dimensional discrete maps.

For example, in Duffing's oscillator, the forcing term is the one containing the cosine function, if we eliminate it by setting $y = 0$ we find the fixed points to be $(0,0), (1,0)$ and $(-1,0)$. Of these, the first is a saddle point and the others are centers in a frictionless system $(\delta =0)$ or drains if the system is dissipative $(\delta > 0)$.