Basic Terms of Nonlinear Dynamics
Nonlinear dynamics is a language to talk about dynamical systems.
Here, brief definitions are given for the basic terms of this language.
All these terms will be illustrated at the pendulum.
- Dynamical system:
- A part of the world which can be seen as a self-contained entity with some temporal
behavior. In nonlinear dynamics, speaking about a dynamical system usually means to
speak about an abstract mathematical system which is a model for such an entity.
Mathematically, a dynamical system is defined by its state and by
A pendulum is an example for a dynamical system.
- A number or a vector (i.e., a list of numbers) defining the state of the dynamical system
uniquely. For the undriven pendulum, the state is uniquely defined by the
angle and the angular velocity := d/dt.. In the case of driving, the driving phase is also needed
because the pendulum becomes a nonautonomous system.
In spatially extended systems, the state is often a field (a scalar
field or a vector field). Mathematically spoken, fields are functions with space
coordinates as independent variables. The velocity field of a fluid is a well-known example.
- Phase space:
- All possible states of the system. Each point in the phase space corresponds to a unique
state. In the case of the undriven pendulum, the phase space has two dimensions
whereas for driven pendula it has three. The dimension of the phase space is
infinite in cases where the system state is defined by a field.
- Dynamics or equation of motion:
- The causal relation between the present state and the next state in the future.
It is a deterministic rule which tells us what happens in the next time step.
In the case of a continuous time, the time step is infinitesimally small.
Thus, the equation of motion is a differential equation or a system of
where u is the state and t is the time variable.
An example is the equation of motion of an undriven and
undamped pendulum. In the case of a discrete time, the time steps are nonzero
and the dynamics is a map:
with the discrete time n. An example is the baker map.
Note, that the corresponding physical time points tn
do not necessarily occur equidistantly. Only the order has to be the same.
That is, n < m implies tn <
The dynamics is linear if the causal relation between the present state and
the next state is linear. Otherwise it is nonlinear.
What happens if the next state isn't uniquely defined by the present one?
In general, this is an indication that the phase space is not complete.
Thus, there are important variables determining the state which had been forgotten.
Of course, this is a crucial point while modelling a real-life systems.
But beside this, there are two important classes of systems where the phase space
is incomplete: The nonautonomuous systems
and the stochastic systems. A nonautonomous systems has an equation of motion
which depends explicitly on time.
Thus, the dynamical rule governing the next state not only depends on the present
state but also at the time it applies.
A driven pendulum is an example of a nonautonomuous system.
Fortunately, there is an easy way to make the phase space complete.
Simply include the time into the definition of the state!
Mathematically, this is done by introducing a new state variable .
Its dynamics reads d/dt = 1, or n+1 = n,
depending on whether time is continuous or discrete.
For the periodically driven pendula, it is also natural to take the
driving phase as the new state variable.
Its equation of motion reads
where f is the driving frequency (the angular driving frequency is
d/dt = 2f,
In a stochastic system, the number and the nature
of the variables necessary to complete the phase space is usually unknown.
Therefore, the next state can not be deduced from the present one.
The deterministic rule is replaced by a stochastic one.
Instead of the next state, it gives only the probabilities of all points in
the phase space to be the next state.
- Orbit or trajectory:
- A solution of the equation of motion. In the case of continuous time,
it is a curve in phase space parametrized by the time variable.
For a discrete system it is an ordered set of points in the phase space.
- The mapping of the whole phase space of a continuous dynamical system onto
itself for a given time step t.
If t is an infinitesimal time step dt, the flow is just given by
the right-hand side of the equation of motion (i.e., F).
In general, the flow for a finite time step is not known analytically because
this would be equivalent to have a solution of the equation of motion.
As an example, the figure shows the flow of a damped pendulum.
The black arrows of the vector field F are tangential at the
In a two-dimensional phase space, you can draw a qualitative
picture of the flow and the orbits. First, draw the so-called null clines.
These are the lines were the time derivative
of one component of the state variable is zero.
Here, one null cline is the angle axes because the time derivative of the angle is
zero when the angular velocity is zero.
The other null cline is = - 02sin/.
On these null clines, draw the vector field vertical or horizontal, respectively.
Between the null clines draw the vector field in the direction north east,
south east, south west, or north west. The direction is determined by the
signs of d/dt and d/dt. At the crossing points of the null clines, the
vector field is zero, i.e., d/dt =
0 and d/dt = 0.
These points are called fixed points.
They correspond to stationary solutions.
Fixed points are examples of non-wandering sets.
They can be either stable or unstable.
- Poincaré section and
A carefully chosen (curved) plane in the phase space that is crossed
by almost all orbits.
It is a tool developed by Henri Poincaré (1854-1912) for a visualization
of the flow in a phase space of more than
two dimensions. The Poincaré section has one dimension less than the phase space.
The Poincaré map maps the points of the Poincaré
section onto itself. It relates two consecutive intersection points.
Note, that only those intersection points counts which come
from the same side of the plane.
A Poincaré map turns a continuous dynamical system
into a discrete one.
If the Poincaré section is carefully chosen no information is lost concerning
the qualitative behavior of the dynamics.
Poincaré maps are invertable maps because one gets un
from un+1 by following the orbit backwards.
In the Pendulum Lab, you can turn the
oscilloscope into a special Poincaré map called
stroboscopic map were the Poincaré section is defined by a certain
phase of the time-periodic driving.
- Non-wandering set:
- A set of points in the phase space having the following property:
All orbits starting from a point of this set come
arbitrarily close and arbitrarily often to any point of the set.
Non-wandering sets are coming in four flavors:
The first three types can also occur in linear dynamics.
The fourth type appears only in nonlinear systems.
Its possibility was first anticipated by the genius of Henri Poincaré (1854-1912).
In the meanwhile, computers had turned this previously counterintuitive behavior into
a widespread experience. In the seventies, this irregular behavior was termed
In the Poincaré map, limit cycles become fixed points.
A non-wandering set can be either stable or unstable.
Changing a parameter of the system can change the stability
of a non-wandering set.
This is accompanied by a change of the number of non-wandering sets due to
- Fixed points:Stationary solutions. For the undriven pendulum and for the
vertically driven pendulum, = 0, d/dt = 0 and = 180°,
d/dt = 0 are fixed points.
- Limit cycles:Periodic solutions. These solutions are common in weakly
- Quasiperiodic orbits: Periodic solutions with at least two incommensurable
frequencies (i.e., the ratio of the frequencies is an irrational number).
In the Pendulum Lab, these solutions occur only in undamped but
- Chaotic orbits: Bound non-periodic solutions.
These solutions occur in driven pendula if the driving is strong enough.
© 1998 Franz-Josef Elmer,
last modified Wednesday, July 22, 1998.