Stability Theory

Stability theory
In mathematics, stability theory addresses the stability of solutions of

differential equations and of trajectories of dynamical systems under
small perturbations of initial conditions. The heat equation, for
example, is a stable partial differential equation because small
perturbations of initial data lead to small variations in temperature at a
later time as a result of the maximum principle. In partial differential
equations one may measure the distances between functions using Lp
norms or the sup norm, while in differential geometry one may
measure the distance between spaces using the Gromov–Hausdorff
distance.
In dynamical systems, an orbit is called Lyapunov stable if the

forward orbit of any point is in a small enough neighborhood or it
stays in a small (but perhaps, larger) neighborhood. Various criteria
have been developed to prove stability or instability of an orbit. Under
favorable circumstances, the question may be reduced to a well-
studied problem involving eigenvalues of matrices. A more general
method involves Lyapunov functions. In practice, any one of a
Stability diagram classifying Poincaré maps of linear autonomous system
number of different stability criteria are applied.
as stable or unstable according to their features. Stability generally increases to the
left of the diagram.[1] Some sink, source or node are equilibrium points.
Contents
Overview in dynamical systems
Stability of fixed points
Maps
Linear autonomous systems
Non-linear autonomous systems
Lyapunov function for general dynamical systems
See also
References
External links
Overview in dynamical systems

Many parts of the qualitative theory of differential equations and dynamical systems deal with asymptotic properties of solutions and the trajectories—what
happens with the system after a long period of time. The simplest kind of behavior is exhibited by equilibrium points, or fixed points, and by periodic
orbits. If a particular orbit is well understood, it is natural to ask next whether a small change in the initial condition will lead to similar behavior. Stability
theory addresses the following questions: Will a nearby orbit indefinitely stay close to a given orbit? Will it converge to the given orbit? In the former case,
the orbit is called stable; in the latter case, it is called asymptotically stable and the given orbit is said to be attracting.
An equilibrium solution to an autonomous system of first order ordinary differential equations is called:
stable if for every (small) , there exists a such that every solution having initial conditions within distance i.e.
of the equilibrium remains within distance i.e. for all .
asymptotically stable if it is stable and, in addition, there exists such that whenever then as
.
Stability means that the trajectories do not change too much under small perturbations. The opposite situation, where a nearby orbit is getting repelled from
the given orbit, is also of interest. In general, perturbing the initial state in some directions results in the trajectory asymptotically approaching the given one
and in other directions to the trajectory getting away from it. There may also be directions for which the behavior of the perturbed orbit is more
complicated (neither converging nor escaping completely), and then stability theory does not give sufficient information about the dynamics.
One of the key ideas in stability theory is that the qualitative behavior of an orbit under perturbations can be analyzed using the linearization of the system
near the orbit. In particular, at each equilibrium of a smooth dynamical system with an n-dimensional phase space, there is a certain n×n matrix A whose
eigenvalues characterize the behavior of the nearby points (Hartman–Grobman theorem). More precisely, if all eigenvalues are negative real numbers or
complex numbers with negative real parts then the point is a stable attracting fixed point, and the nearby points converge to it at an exponential rate, cf
Lyapunov stability and exponential stability. If none of the eigenvalues are purely imaginary (or zero) then the attracting and repelling directions are
related to the eigenspaces of the matrix A with eigenvalues whose real part is negative and, respectively, positive. Analogous statements are known for
perturbations of more complicated orbits.
Stability of fixed points

The simplest kind of an orbit is a fixed point, or an equilibrium. If a mechanical system is in a stable equilibrium state then a small push will result in a
localized motion, for example, small oscillations as in the case of a pendulum. In a system with damping, a stable equilibrium state is moreover
asymptotically stable. On the other hand, for an unstable equilibrium, such as a ball resting on a top of a hill, certain small pushes will result in a motion
with a large amplitude that may or may not converge to the original state.
There are useful tests of stability for the case of a linear system. Stability of a nonlinear system can often be inferred from the stability of its linearization.
Maps
Let f: R → R be a continuously differentiable function with a fixed point a , f(a) = a . Consider the dynamical system obtained by iterating the function
f:
The fixed point a is stable if the absolute value of the derivative of f at a is strictly less than 1, and unstable if it is strictly greater than 1. This is because
near the point a , the function f has a linear approximation with slope f'(a):
Thus
which means that the derivative measures the rate at which the successive iterates approach the fixed point a or diverge from it. If the derivative at a is
exactly 1 or −1, then more information is needed in order to decide stability.
There is an analogous criterion for a continuously differentiable map f: Rn → Rn with a fixed point a , expressed in terms of its Jacobian matrix at a ,
Ja(f). If all eigenvalues of J are real or complex numbers with absolute value strictly less than 1 then a is a stable fixed point; if at least one of them has
absolute value strictly greater than 1 then a is unstable. Just as for n =1, the case of the largest absolute value being 1 needs to be investigated further —
the Jacobian matrix test is inconclusive. The same criterion holds more generally for diffeomorphisms of a smooth manifold.
Linear autonomous systems
The stability of fixed points of a system of constant coefficient linear differential equations of first order can be analyzed using the eigenvalues of the
corresponding matrix.
An autonomous system
where x(t) ∈ Rn and A is an n×n matrix with real entries, has a constant solution
(In a different language, the origin 0 ∈ Rn is an equilibrium point of the corresponding dynamical system.) This solution is asymptotically stable as
t → ∞ ("in the future") if and only if for all eigenvalues λ of A, Re(λ) < 0. Similarly, it is asymptotically stable as t → −∞ ("in the past") if and only if
for all eigenvalues λ of A, Re(λ) > 0 . If there exists an eigenvalue λ of A with Re(λ) > 0 then the solution is unstable for t → ∞ .
Application of this result in practice, in order to decide the stability of the origin for a linear system, is facilitated by the Routh–Hurwitz stability criterion.
The eigenvalues of a matrix are the roots of its characteristic polynomial. A polynomial in one variable with real coefficients is called a Hurwitz
polynomial if the real parts of all roots are strictly negative. The Routh–Hurwitz theorem implies a characterization of Hurwitz polynomials by means of
an algorithm that avoids computing the roots.
Non-linear autonomous systems
Asymptotic stability of fixed points of a non-linear system can often be established using the Hartman–Grobman theorem.
Suppose that v is a C1-vector field in Rn which vanishes at a point p , v(p) = 0 . Then the corresponding autonomous system
has a constant solution
Let Jp(v) be the n×n Jacobian matrix of the vector field v at the point p . If all eigenvalues of J have strictly negative real part then the solution is
asymptotically stable. This condition can be tested using the Routh–Hurwitz criterion.
Lyapunov function for general dynamical systems

A general way to establish Lyapunov stability or asymptotic stability of a dynamical system is by means of Lyapunov functions.
See also
Asymptotic stability
Hyperstability
Linear stability
Orbital stability
Stability criterion
Stability radius
Structural stability
von Neumann stability analysis
References
1. Egwald Mathematics - Linear Algebra: Systems of Linear Differential Equations: Linear Stability Analysis (http://www.egwald.ca/linear
algebra/lineardifferentialequationsstabilityanalysis.php) Accessed 10 October 2019.
Philip Holmes and Eric T. Shea-Brown (ed.). "Stability" (http://www.scholarpedia.org/article/Stability). Scholarpedia.
External links
Stable Equilibria (http://demonstrations.wolfram.com/StableEquilibria/) by Michael Schreiber, The Wolfram Demonstrations Project.
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Stability Theory

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Stability theory

In mathematics, stability theory addresses the stability of solutions of

In dynamical systems, an orbit is called Lyapunov stable if the

Overview in dynamical systems

Stability of fixed points

Linear autonomous systems

Non-linear autonomous systems

has a constant solution

Lyapunov function for general dynamical systems

Philip Holmes and Eric T. Shea-Brown (ed.). "Stability" (http://www.scholarpedia.org/article/Stability). Scholarpedia.

Retrieved from "https://en.wikipedia.org/w/index.php?title=Stability_theory&oldid=1010008143"

This page was last edited on 3 March 2021, at 10:30 (UTC).

Text is available under the Creative Commons Attribution-ShareAlike License;

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