Stability theory

The mathematical stability theory is concerned with the development of disorders that occur as a deviation from certain states of dynamical systems ( stability). Such a condition may for example be a rest position or a certain orbit, for example, a periodic orbit.

In addition to its theoretical importance, the stability theory is in physics and theoretical biology as well as in technical fields, for example in engineering mechanics or control theory.

The solutions to the problems of stability theory are ordinary and partial differential equations.

Mathematical stability concepts

For the characterization of the stability of the tranquility of a dynamic system several stability concepts exist, each with slightly different statement:

  • A rest position is called Lyapunov stable if a sufficiently small perturbation also always remains small. More precisely: for each one exists such that is valid for all times and all trajectories.
  • A rest position is called attractive if there is a such that each trajectory exists with for all and fulfills the following limit condition:
  • A rest position is called asymptotically stable if it is Lyapunov stable and attractive.
  • A rest position is called neutral stable or marginally stable if it is stable, but not asymptotically stable.

For the case of discrete systems, which are described by differential equations, the rest is the same fixed point of the recurrence and are similar stability definitions common.

Linear time-invariant systems

For linear time-invariant systems, the stability of the transfer function by the position of the poles can be read in the s-plane:

  • Asymptotic stability: if all poles lie in the left half s- plane,
  • Cross Stability: when no pole is located in the right half s- plane and at least a simple pole, but not a multiple pole, located on the imaginary axis of the s- half-plane,
  • Instability: otherwise (if at least one pole is located in the right half s- plane, or when at least a multiple pole on the imaginary axis of the s- plane ).

Direct method of Lyapunov and Lyapunov function

Lyapunov developed in 1883 the so-called Direct or Second method ( the first method was the linearization, see below) to verify the above stability properties of concrete systems. For this purpose, first we define a dynamic system of the form and a real-valued differentiable function, the orbital derivative

A real-valued differentiable function is called Lyapunov function ( for the vector field ) if the following holds for all points of the phase space. A Lyapunov function is a quite powerful tool for a stability proof, as the following two criteria show:

The use of a Lyapunov function ( ie without that one would have to solve the differential equation ) is called direct method, because it means directly from the vector field without knowledge of the trajectories can gain information on the stability of a rest position.

Ljapunowgleichung

For the case of linear systems, for example, can always be used as a Lyapunov function is a positive definite quadratic form. You obviously satisfies the above conditions (1) and (2). Condition (3) leads to the Lyapunov equation

.

If is positive definite, then is a Lyapunov function. For stable linear systems can always find such a function.

Stability analysis of linear and non-linear systems

A dynamic system is represented by the equation. We consider a problem at the time. If the system is linear, this error can be fully expressed by the Jacobian matrix of the first derivatives with respect to. It is not linear, it can be " linearised " ie the function according to the Taylor development, if the error is small enough. In both cases, is obtained for the time evolution of

Said Jacobian matrix at the site of the rest position. This development will therefore be largely determined by the eigenvalues ​​of the Jacobian matrix. Specifically, there are three different cases:

Examples

An investigated deformation state of strength of materials, or a state of motion dynamics can switch to a different state from a to be determined stability limit. Associated with this are usually not linearly increasing deformations or movements that can lead to the destruction of structures. To avoid this, knowing the limit of stability is an important criterion for the design of components.

Other examples are:

  • Stability testing of control loops,
  • Dynamics of insect populations
  • Euler Buckling,
  • Tilting of slender beams,
  • Growth of small disturbances in a boundary layer, leading to the laminar- turbulent transition.
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