In the previous post in the series, we looked at the Dynamical systems. These systems are nonlinear and cannot be analyzed easily, the difficulty arising from the fact that most of them aren’t exactly solvable. By exactly solvable, we mean a system that can be solved to get solutions in ‘closed’ algebraic form or a ‘formula’ in which you input initial conditions to predict the future state. In nonlinear dynamical systems, initial conditions themselves dictate how the system will behave, preventing us from getting an algebraic solution. Slightly different initial conditions can lead to very different final states of the system, even if the system is deterministic.

However, the situation isn’t helpless. The key to analyzing these systems is to understand that knowing the solutions isn’t necessarily the only way to analyze the system. If you are only interested in knowing what the solutions look like, you need not know the exact solutions. Nonlinear Dynamics (NLD) is the branch of Physics/Mathematics that deals with such systems. The approach in NLD is more geometrical than analytical. Instead of finding the exact solutions, we look at how they behave in the state space and try to visualize them geometrically. However, let’s define what we mean by phase space and state space first.

State Space

The state space or phase space, as physicists like to call it, is an abstract n-dimensional construct in which there are twice as many axes as there are degrees of freedom in the system. For example, if a particle is moving in a straight line, there is only one direction (along the line, say x -axis) for it to move. It can, however, change its velocity. In this case, traditionally we would call it a one-dimensional system. The corresponding phase space would then be two-dimensional, one for position coordinate (x) and one for momentum/velocity ( \dot{x} or p). Physicists, in general, tend to prefer momentum as the other coordinate. We, however, would prefer \dot{x} or generalized velocity.

By a one dimensional system, we generally mean that the system is allowed to move in only one spatial direction. e.g.

    \[\dot{x} = cos 6x\]

There is only one variable x here. In this case, the configuration space would be just a straight line (x axis).

However, the phase space would be two dimensional, consisting of x and \dot{x} as the corresponding axes.

Figure: cos 6x versus x. This is the phase space, not the actual trajectory.

If the system has more than 2 degrees of freedom, the phase space becomes almost impossible to visualize. We have to look for a way to reduce the dimensions of the system in such a way that the ‘reduced’ space preserves the nature of underlying dynamics but is of lower dimensions than the original phase space. We can achieve this by means of surface of section or Poincare maps. But we’re getting ahead of ourselves. In the next post, we shall see how phase space description helps us see things visually, by means of what we call as ‘phase flow’.

Reference: Strogatz, Steven H. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. CRC Press, 2018.

Nonlinear Dynamics: Phase Space

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