Written by Alexei Gilchrist, updated some time ago
The map of phase space of a dynamical system is a really convenient way of summarising the behaviour of a system. In particular it doesn't require a solution to the equations of motion.
Level: 2, Subjects: Oscillators

1 Introduction

There a neat way of visualising the dynamics of small-dimension deterministic systems, especially nonlinear dissipative systems where we may have no hope of finding closed form solutions. Recall that in all the solutions to one-dimensional systems we have found there are two free parameters to be determined by the initial conditions. In effect the freedom we have is in specifying the initial position and velocity of the system. This is a generic feature of second order differential equations. Since these two quantities are independent conditions we can plot them against each other — a point on this graph will represent a particular starting value and it will determine the motion for all future times. That is, from that point on we can then track the evolution of the systems as a path or trajectory through the space of states. This graph is a plot of the phase-space of the system (well technically it should be a co-ordinate vs the canonical momentum for that co-ordinate but velocity will do for our purposes).

With this new tool at our disposal we could sample the phase space and draw a small arrow at each point indicating the direction the system's state would take if it where to have that initial value, and in this way we can draw a `map' of the dynamics and see how the system would behave with any initial condition at a glance!

In practice though, we can just map out some key features in this space, such as the location of fixed points (starting points where the system does not change in time), the general flow of trajectories in different regions, and the curves that separate different flows which are called a separatrix. This map would then give a very good idea of the dynamics without ever having solved the equations. It should also be easy to see how to extend this description to the multidimensional phase spaces describing more complex systems.

2 Damped harmonic oscillator

As a first example we will determine the phase space of a damped harmonic oscillator (see damped-harmonic-oscillator) for which we already know an explicit general solution. The equation of motion for this system is \begin{equation}\label{eq:damped-eqn-motion} \ddot{x} + \gamma \dot{x} + \omega_0^2 x = 0. \end{equation}

The general solution to a damped harmonic oscillator is \begin{equation*} x(t) = e^{-\frac{\gamma}{2} t} \left[x_0 \cosh(i\omega_d t)-\frac{i(2v_0+\gamma x_0)}{2 \omega_d} \sinh(i \omega_d t) \right]. \end{equation*} where \(x_0\) and \(v_0\) are the initial position and velocity respectively. If we differentiate this expression with respect to time we get \begin{equation*} v(t) = e^{-\frac{\gamma}{2} t} \left[v_0 \cosh(i\omega_d t)+i\left(x_0 \omega_d+\frac{\gamma(2v_0+\gamma x_0)}{4 \omega_d}\right)\sinh(i \omega_d t) \right]. \end{equation*} These are messy looking expressions but if we plot one against the other for some initial value the result is surprisingly simple

The damped oscillator for \(\gamma=0.3\) and \(\omega_d=2\), starting from \(x_0=1\) and \(v_0=0\). The dashed curves are just for reference. Notice that the phasor shortens the most when the velocity is largest. This is a consequence of the velocity dependent damping in the model.
This plot should be familiar from the phasor diagram for sinusoid motion, the only difference here is that the shape is slightly squashed in one direction (see caption for explanation), and the phasor is slowly getting shorter. The complicated looking solutions above are the \(x\) and \(v\) projections of this curve.

What we would really like to do is be able to determine this motion without solving the equation of motion. Luckily there is a technique for doing just this. In order to draw an arrow pointing in the direction of the flow at some point \((x,v)\) in phase space, we need to know the direction of the change in time at that point. In other words, we need to know the time derivatives \((\dot{x},\dot{v})\).

It's always possible to reduce an \(n^\mathrm{th}\) order differential equation to \(n\) coupled first order differential equations by introducing extra variables. In the case of \eqref{eq:damped-eqn-motion} an obvious new variable to introduce is the velocity: \(\dot{x}=v\). Then rewriting the remainder to use \(v\) gives two coupled first order equations, \begin{align*} \dot{x} &= v \\ \dot{v} &= -\gamma v - \omega_0^2 x, \end{align*} and tells us the time derivatives for \(x\) and \(v\) which we wanted.

The first thing we should do is find any fixed points for the system. By definition these points don't change with time so \(\dot{x}=\dot{v}=0\). The two equations above imply that \(v=0\) and consequently \(x=0\). As this is the only solution, the point \((0,0)\) is the only fixed point.

To figure out the direction of flow at a particular point we just use the coupled differential equations. Say we want to know in which direction a trajectory at the position \((x_0,0)\) will move. Plugging these values in gives \begin{align*} \dot{x} &= 0 \\ \dot{v} &= - \omega_0^2 x_0. \end{align*} That is, there is no change in the \(x\) direction so the arrow will be vertical in the diagram and in the negative \(v\) direction if \(x_0\) is positive and vice-versa. For the point \((0,v_0)\) we get \begin{align*} \dot{x} &= v_0 \\ \dot{v} &= -\gamma v_0, \end{align*} so the direction is \(+x\) and \(-v\) if \(v_0\) is positive or \(-x\) and \(+v\) if \(v_0\) is negative. In either case the change in \(v\) is a factor \(\gamma\) of the change in \(x\). These diagrams are most easily constructed by using a computer algebra system such as Wolfram Alpha and the command StreamPlot.

Flow of trajectories for a damped harmonic oscillator. Shown are maps for \(\omega_0=2\), with \(\gamma=1\) (left) and \(\gamma=4\) (right). The right plot corresponds to a critically-damped system.

You can explore this map below.

Click figure to download the CDF demo.
The dynamical map of the damped harmonic oscillator. Pick some parameters below and explore how the map changes. Also by clicking somewhere on the map you can see the trajectory the system would take if it started from that point. Play with over and under damping.

3 Damped Pendulum

Consider now the equation of motion for a damped rigid pendulum (pendulum): \begin{equation*} \ddot{\theta} + \gamma\dot{\theta} + \omega_0^2\sin(\theta) = 0. \end{equation*} This model has angular velocity dependent damping at a rate \(\gamma\).

Following the prescription above, we can introduce a new variable \(v=\dot{\theta}\), then split the equation into coupled differential equations \begin{align*} \dot{\theta} &= v \\ \dot{v} &= -\gamma v - \omega_0^2\sin(\theta). \end{align*} Clearly if \(\theta\) is small enough then \(\sin\theta\approx\theta\) and we have the same set of equations as for the damped harmonic oscillator. For larger \(\theta\) the phase space differs. Note that the phase space in this case is really a cylinder as the angle wraps around after \(2\pi\) radians.

Setting \(\dot{\theta}=\dot{v}=0\) we find two fixed points at \((\theta=0,v=0)\), and \((\theta=\pi,v=0)\). The first corresponds to the pendulum hanging straight down and the second the the pendulum balanced pointing straight up. With any amount of damping (\(\gamma\ge 0\)) all the trajectories flow into the first fixed point.

Click figure to download the CDF demo.
The dynamical map of the damped pendulum. Note that \(\theta\) wraps around in \(2\pi\) radians so that for instance the points indicated by the dashed lines are the same. The are two fixed points in the system, the green point is stable in the sense that a small perturbation away from the point will get drawn back, the red point is unstable.

Exercises