My aim in this series of posts is to plot a simple course through elementary dynamical systems theory to arrive quickly at some of the basic elements which appear in physical theories like quantum mechanics. In learning physics we normally encounter many of these concepts only when we first need them, and rarely see clearly how generic they turn out to be. It should help to follow the story from the beginning—all we need is first-order ordinary differential equations.

Table of Contents

Direct Integration

Our subject will be the trajectory of a single point over time, for now in or , in response to a velocity function

In the simplest case of a velocity depending on time only, , the the trajectory is given by the integral

if the velocity is constant the solution is simple:

where includes the initial condition as a “parameter”.

For non-constant velocity we add the integral:

which simply represents the repeated application of the linearization in the limit .

For smooth , the integral expression must be equivalent to a Taylor Series where hardcodes the first derivative, the second, etc.:

We could generate the individual terms of this series by repeatedly expanding the previous integrand with the Fundamental Theorem of Calculus:

This series is suggestive of the series expansion of an exponential , where is a shorthand for . This “time-translation operator”, applied to the function and evaluated at , is

This expression is basically a general solution to time-translation, and could be evaluated term-by-term by substituting .

The system specified by a velocity function is fairly trivial. Solving it amounts to mere integration. If we know and all its higher derivatives, the entire trajectory is determined by the initial values.

Generating a Flow

What if varies with space as ? Here plays the role of a “velocity field” indicating the direction of motion at any point. Unlike a real physical system, there is no “momentum” or “inertia”; no tendency to keep going in a direction. (Such a first-order system without inertia could represent the motion of a particle in an extremely viscous fluid which dampens all momentum. And a system with inertia could be modeled this way if we treated and as a two-dimensional, first-order system, which we’ll do later on.)

The ODE

can be integrated to an awkward but general solution:

For example, if then the solution is

which is readily solved (ignoring the absolute values, which add a wrinkle) to give

representing a trajectory which exponentially grows or decays, depending on the sign of .

Calling the l.h.s. integral , we see that time evolution amounts to a translation in ,

If is simple enough to be invertible, this has an exact solution:

This is not as nice an expression as the others, but it looks like something we can expand in a series. The inverse function theorem for the derivative will be useful: from the fact that we can evaluate the successive derivatives of ,

Evidently each derivative applies a factor of to the previous one.

With these, the series expansion is

and the last line looks like the series expansion of an exponential function whose argument is the “generator” :

We might have been able to guess the form of the generator by observing that

which is suggestive of an ODE with solution . But this feels sketchy—we seem to be blurring the idea of what actually is here. We’ll come back to this.

The strange-looking exponential here is really an “operator” acting on functions ,

and the time-evolution of the identity function would give us itself

This notation now looks a little odd, because is now only “bound” within the function . We can clean it up somewhat by writing as or just , understanding it to act on the spatial coordinate to its right in either case, and writing as :

Higher Dimensions

Next we ought to visit the -dimensional case of , to see which features are generic to multi-dimensional systems. We won’t need a time-dependent , so I’ll put that off until later. This will also provides some examples of what such an exponential operator can do.

What happens? Well, nothing about the derivation in the previous section required one dimension, so we can quote the solution in the function representation:

and the time evolution of a trajectory is the same expression on the identity :

Note we’ve promoted to , with a dot product.

Of interest is the simple case , a pure matrix multiplication, which will serve as a prototype for the local linearization of a generic . Here

and

so the general solution is simply

The corresponding 1D case had an exact solution , i.e. exponential growth or decay. With more dimensions, much more can happen:

Exponential growth and decay, at once along different eigenvectors, e.g.:

Rotations, where two dimensions flow into each other while other dimensions do other things:

where

Shears, the simplest case of which is a nilpotent matrix leading to linear trajectories of varying speeds:

A general nilpotent matrix, for which for some power , will lead to some polynomial growth in time, since the series

terminates at the th term.

Transients of various kinds:

A general real matrix could shear any “subsystem” into any other one, with various results, e.g.:

  • The dimension being sheared-into could have its own growth or decay rate:

Here, if , the resulting shear would be a transient effect which eventually dies off, while it would boost the already-unbounded growth of .

  • If is a pure rotation/oscillation, while is also sheared into , then will couple to the oscillation at a lag:
  • If is exponentially decaying, while also shearing into , then will grow transiently before leveling off:
  • If multiple dimensions could shear into the same one, various combinations of transient effects could occur, e.g.: a growth followed by decay, decay followed by growth, initial oscillations drowned out by growth, etc.

In general the classification scheme is:

  • if the matrix is normal, , then its eigenvectors are orthogonal, and its dynamics “factor” into distinct orthogonal dimensions with their own growth or decay, or into pairs of dimensions exhibiting rotations. Classifying eigenvalues suffices to characterize behavior.
  • if not, the eigenvectors are non-orthogonal and can feed transiently into each other.
  • if furthermore the matrix is “defective”—it has a non-trivial Jordan Normal form, and fewer eigenvectors than dimensions—then polynomial-in-time terms like appear.

Real physical systems rarely exhibit these latter exotic behaviors for various reasons—among them that transient growths tend to violate conservation laws, and that nilpotence as a property is unstable w.r.t. perturbations of . But these systems are still interesting as models of reality, because they tend to lie on the “phase change” barriers of dynamical systems, e.g. near resonances, or near the limitations of applicability of a particular model.



I find it clarifying to approach the analysis of linear systems with the view of time-evolution as an exponential already in hand. Exponential solutions are generic because exponentiation is generic—it represents the basic feedback loop by which a trajectory at later times experiences the compounding effects of earlier times.

Of course, in full generality we would have to apply that entire analysis to the local linearization of near a point :

A careful analysis might proceed first by identifying the fixed points , linearizing in their local neighborhoods to determine stability, and then dividing the overall space into regions according to which fixed points or limiting behavior each flows into.

But “fixed points” hardly tell the whole story. The following system, for example, exhibits a “limit cycle”: a circle of fixed-points were , with trajectories on either side of the circle flowing stably towards it,

The behavior which is easier to see in polar coordinates:

We can read off the behavior:

  • at ,
  • grows when and shrinks when , approaching the fixed point from both sides.
  • circulates at a constant rate all the while.
  • is an unstable fixed point.


We’ll stop there for now. The next post will take on the question of what exactly we’re talking about, towards the general case of a time-varying velocity field .