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 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.
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.
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.
to an autonomous system of first order ordinary differential equations is called: Stability means that the trajectories do not change too much under small perturbations.
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.
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.
The paradigmatic case is the stability of the origin under the linear autonomous differential equation
, we first find the Jordan normal form of the matrix, to obtain a basis
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.
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.
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.
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.)
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.
Asymptotic stability of fixed points of a non-linear system can often be established using the Hartman–Grobman theorem.
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.
