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
Overview in dynamical systems
Many parts of the
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
Stability of fixed points in 2D
The paradigmatic case is the stability of the origin under the linear autonomous differential equation where and is a 2-by-2 matrix.
We would sometimes perform change-of-basis by for some invertible matrix , which gives . We say is " in the new basis". Since and , we can classify the stability of origin using and , while freely using change-of-basis.
Classification of stability types
If , then the rank of is zero or one.
- If the rank is zero, then , and there is no flow.
- If the rank is one, then and are both one-dimensional.
- If , then let span , and let be a preimage of , then in basis, , and so the flow is a shearing along the direction. In this case, .
- If , then let span and let span , then in basis, for some nonzero real number .
- If , then it is unstable, diverging at a rate of from along parallel translates of .
- If , then it is stable, converging at a rate of to along parallel translates of .
If , we first find the Jordan normal form of the matrix, to obtain a basis in which is one of three possible forms:
- where .
- If , then . The origin is a source, with integral curves of form
- Similarly for . The origin is a sink.
- If or , then , and the origin is a saddle point. with integral curves of form .
- where . This can be further simplified by a change-of-basis with , after which . We can explicitly solve for with . The solution is with . This case is called the "degenerate node". The integral curves in this basis are central dilations of , plus the x-axis.
- If , then the origin is an degenerate source. Otherwise it is a degenerate sink.
- In both cases,
- where . In this case, .
- If , then this is a spiral sink. In this case, . The integral lines are logarithmic spirals.
- If , then this is a spiral source. In this case, . The integral lines are logarithmic spirals.
- If , then this is a rotation ("neutral stability") at a rate of , moving neither towards nor away from origin. In this case, . The integral lines are circles.
The summary is shown in the stability diagram on the right. In each case, except the case of , the values allows unique classification of the type of flow.
For the special case of , there are two cases that cannot be distinguished by . In both cases, has only one eigenvalue, with algebraic multiplicity 2.
- If the eigenvalue has a two-dimensional eigenspace (geometric multiplicity2), then the system is a central node (sometimes called a "star", or "dicritical node") which is either a source (when ) or a sink (when ).[2]
- If it has a one-dimensional eigenspace (geometric multiplicity1), then the system is a degenerate node (if ) or a shearing flow (if ).
Area-preserving flow
When , we have , so the flow is area-preserving. In this case, the type of flow is classified by .
- If , then it is a rotation ("neutral stability") around the origin.
- If , then it is a shearing flow.
- If , then the origin is a saddle point.
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
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
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
Linear autonomous systems
The stability of fixed points of a system of constant coefficient
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,
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
- Chaos theory
- Lyapunov stability
- Hyperstability
- Linear stability
- Orbital stability
- Stability criterion
- Stability radius
- Structural stability
- von Neumann stability analysis
References
- ^ Egwald Mathematics - Linear Algebra: Systems of Linear Differential Equations: Linear Stability Analysis Accessed 10 October 2019.
- ^ "Node - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2023-03-30.
- Philip Holmes and Eric T. Shea-Brown (ed.). "Stability". Scholarpedia.
External links
- Stable Equilibria by Michael Schreiber, The Wolfram Demonstrations Project.