Dynamical system

An (deterministic ) mathematical model is a dynamic system of a time-dependent process, which is homogeneous with respect to the time, thus its further course only from the initial state, but does not depend on the choice of the initial time point. The concept of the dynamic system is in its present form back to the mathematician Henri Poincaré and George David Birkhoff.

Dynamical systems have many uses in processes in everyday life and provide insights into many areas not only of mathematics (eg, number theory, stochastics ), but also in physics (eg, pendulum movement, climate models ) or theoretical biology (eg. predator-prey models).

We distinguish between discrete and continuous time evolution. In a discrete-time dynamic system, the states change in equidistant time steps, i.e., in successive time intervals is always the same size, while during the state changes in a continuous-time dynamic system in infinitesimally small time intervals. The most important means of description for continuous-time dynamical systems are autonomous ordinary differential equations. A mixed system of continuous and discrete subsystems with continuously - discrete dynamics is also referred to as hybrid. Examples of such hybrid dynamics can be found in process engineering (eg dosage feed systems).

Important issues related to dynamic systems relate primarily to their long-term behavior (for example, stability, periodicity, chaos and ergodicity ), the system identification and their control.

  • 2.1 Remarks
  • 3.1 Ordinary Differential Equations
  • 3.2 iteration

Introductory Examples

Exponential Growth

A simple example of a dynamic system, the time evolution of a variable which is subject to an exponential growth, such as a population of a bacteria culture growing unhindered. The state for a fixed time, here by a non-negative real number, namely, if the cumulative value of the population, i.e., the state space of the system is the set of non-negative real numbers. Considering first, the states of the discrete points in time, ie the period of time, applies a constant growth factor. For the condition at a time, this results.

The characterizing feature of a dynamic system is that, although the state is dependent on the elapsed time and the initial value, but not by the choice of the initial time point. Is given as a further exponentially growing population of the same growth factor, but with the initial value. Applies at a time then

The second population that is growing in the same way as the first time period in the time period. This behavior can be still expressed differently: the so-called flow function, the state allocates any time and any initial state at the time, so here satisfied for all and all, the equation

This is called the semigroup property of the flow of a dynamic system.

Spring pendulum

Another source of dynamic systems is the mathematical modeling of mechanical systems, in the simplest case, the motion of a particle under the influence of a force which depends on the location and the speed of, but not specifically to time. The state of such a system at a time is given by the ordered pair, consisting of the location and speed. In particular, the entire movement is uniquely determined by specifying an initial position together with an initial speed. In the case of one-dimensional motion is thus the state space.

A specific example of a pendulum spring should be considered, on the mass element to the ground, the restoring force of the spring as well as possibly applied a speed-dependent friction force. If the total force, the result for the state, the ordinary differential equation system

Wherein the point of the tag, the derivative with respect to time respectively. The first equation states that the velocity is the derivative of position with respect to time, and the second follows directly from Newton's second axiom, according to the mass times acceleration is equal to the force acting on the mass point total force.

It can be shown that even with this system, the flow

Satisfies the semigroup property. If the course of the system state in the state space, ie the so-called web so results in a damped oscillation of the spring pendulum a trajectory that converges spirally to the rest position.


A dynamical system is a triple consisting of a set of true or the period of a non-empty set, the state space, and an operation of on such that for all states and all time points:

If or, then called discrete-time or short discreet, and with or called time or continuously. is also referred to as a discrete or continuous dynamical system for real -time or as invertible or if applicable.

For each ie the figure of the movement and the amount is called by the train or the ( full ) orbit. The positive half orbit or forward orbit is and if is invertible, is the negative semitrajectory or reverse orbit.

A discrete dynamical system is continuous if its state space is a ( non-empty ) metric space and if each belonging to a time transformation is continuous. A continuous dynamical system is called a semi- continuous or flow, when its state space is a metric space, and where each associated with a point in time as well as any transformation of a motion state is continuous. In addition, it is called a continuous -discrete dynamic system, a cascade and a half river a river. The state space of a continuous dynamical system is also called the phase space and from each of the orbit as a curve or trajectory of the phase, which is simply written.

If you couple continuous and given case, additional discrete dynamical systems in a single system, so you call this a continuous - discrete or hybrid dynamic system.


  • In the literature it is often no distinction between dynamical systems and continuous dynamical systems or rivers, also is meant by a river often a differentiable flow ( see below). There are also more general definitions of continuous dynamic systems in which, for example, as a phase space is a topological manifold, a (possibly compact ) Hausdorff space or even just a topological space is taken.
  • In place of the link operation, as in the definition above are often dynamic systems with a legal operation to define the order of the arguments then rotates accordingly.
  • In the definition of the identity property is required by the operation because each state so long no time elapses (ie, for ) not to change. This property means that the belonging to transformation is the identity map on:
  • The semigroup property makes the dynamic system with respect to time homogeneous initially One arrives in time units of the state to state, and then from there to the one is in units of time in units of time on the condition, i.e., the same state as the state directly. The all time points belonging transformations form a commutative semigroup with the composition as a link, and with a neutral element, also the picture is a Halbgruppenhomomorphismus: for all this transformation semigroup with invertible dynamical systems even a group, because for all is the inverse element to
  • A dynamic system or can be accurately then continue to an invertible dynamic system, when the to belonging transformation has an inverse function. There are now and then recursively for all is continuous, so are through for all and also given all belonging to negative times transformations clearly. With so exactly one operation of on explained so that the invertible Continued from is.
  • Because of the semi- group property can be any discrete dynamical system or interpreted as iterative application of belonging to transform with the times as Iterationsindizes: for all and is in addition to all therefore is already determined by clear and can be written easily.
  • Restricts you in a continuous dynamic system, the time to a, then results with always a discrete dynamical system. This discretization is the one in the numerics a great application, such as in the backward analysis. On the other hand, there are natural and technical systems that can be characterized by discontinuous changes in state and modeled in a direct way by discrete dynamical systems.
  • Differentiable (semi) flows are ( semi-) streams, in which each associated with a time transformation is differentiable. In particular, each of these transformations of a differentiable flow is a diffeomorphism.
  • In the theory of dynamical systems are particularly interested in the behavior of trajectories. Here are limit sets and their stability is of great importance. The simplest limit sets are fixed points, which are those points with for all, that those states whose car is the one-element set. Next we are interested for points whose path converges to a fixed point. The main limit sets are fixed points besides the periodic orbits. But it is precisely in nonlinear systems you can meet the complex non-periodic boundary quantities. In the theory of non-linear systems are fixed points, periodic orbits, and general non-periodic boundary quantities under the generic term attractor (or repeller, if repulsive, see also strange attractor ) subsumed. These are examined in detail in the chaos theory.

Important special cases

Ordinary Differential Equations

Continuous dynamic systems occur with ordinary differential equations, especially in the context. Consider the autonomous differential equation

With a vector field in a field. If the equation for all initial values ​​of a for all defined, unique solution has to, then with a continuous dynamic system. The orbits of the system are thus the solution curves of the differential equation. The fixed points are the ones with here; they are also called stationary or critical points of the vector field.


Discrete dynamical systems are iteration of functions in close relationship. If a self-map of an arbitrary set, ie a function that assigns to each again an element, then you can consider the initial value to a recursively defined sequence for. With the sequential execution times ( time ) then applies. The equation shows that this is provided with a discrete dynamic system. Conversely, for a dynamical system by a mapping defined with. The fixed points of such a system are associated with.