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Inmathematics,stability theory addresses the stability of solutions ofdifferential equations and of trajectories ofdynamical systems under small perturbations of initial conditions. Theheat 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 themaximum principle. In partial differential equations one may measure the distances between functions usingL^p norms or the sup norm, while in differential geometry one may measure the distance between spaces using theGromov–Hausdorff distance.

In dynamical systems, anorbit is calledLyapunov 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 involvingeigenvalues ofmatrices. A more general method involvesLyapunov functions. In practice, any one of a number of differentstability criteria are applied.

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Many parts of thequalitative 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 byequilibrium points, or fixed points, and byperiodic 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 calledstable; in the latter case, it is calledasymptotically stable and the given orbit is said to beattracting.

An equilibrium solution${\displaystyle f_e}$ to an autonomous system of first order ordinary differential equations is called:

• stable if for every (small)${\displaystyle ϵ>0}$, there exists a${\displaystyle \delta >0}$ such that every solution${\displaystyle f(t)}$ having initial conditions within distance${\displaystyle \delta }$ i.e. of the equilibrium remains within distance${\displaystyle ϵ}$ i.e. for all${\displaystyle t\ge t_0}$.
• asymptotically stable if it is stable and, in addition, there exists${\displaystyle {\delta }_0>0}$ such that whenever then${\displaystyle f(t)\to f_e}$as${\displaystyle t\to \mathrm{\infty }}$.

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 thelinearization of the system near the orbit. In particular, at each equilibrium of a smooth dynamical system with ann-dimensionalphase space, there is a certainn×n matrixA whoseeigenvalues characterize the behavior of the nearby points (Hartman–Grobman theorem). More precisely, if all eigenvalues are negativereal numbers orcomplex numbers with negative real parts then the point is a stable attracting fixed point, and the nearby points converge to it at anexponential rate, cfLyapunov stability andexponential stability. If none of the eigenvalues are purely imaginary (or zero) then the attracting and repelling directions are related to the eigenspaces of the matrixA with eigenvalues whose real part is negative and, respectively, positive. Analogous statements are known for perturbations of more complicated orbits.

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The paradigmatic case is the stability of the origin under the linear autonomous differential equation${\displaystyle \dot{X}=AX}$ where${\displaystyle X=\left[\begin{array}{c}x\\ y\end{array}\right]}$ and${\displaystyle A}$ is a 2×2 matrix.

We would sometimes perform change-of-basis by${\displaystyle X^{\prime }=CX}$ for some invertible matrix${\displaystyle C}$, which gives${\displaystyle {\dot{X}}^{\prime }=C^{-1}AC{X}^{\prime }}$. We say${\displaystyle C^{-1}AC}$ is "${\displaystyle A}$ in the new basis". Since${\displaystyle detA=det{C}^{-1}AC}$ and${\displaystyle \mathrm{tr}A=\mathrm{tr}{C}^{-1}AC}$, we can classify the stability of origin using${\displaystyle detA}$ and${\displaystyle \mathrm{tr}A}$, while freely using change-of-basis.

If${\displaystyle detA=0}$, then the rank of${\displaystyle A}$ is zero or one.

• If the rank is zero, then${\displaystyle A=0}$, and there is no flow.
• If the rank is one, then${\displaystyle \ker A}$ and${\displaystyle \mathrm{im}A}$ are both one-dimensional.
  • If${\displaystyle \ker A=\mathrm{im}A}$, then let${\displaystyle v}$ span${\displaystyle \ker A}$, and let${\displaystyle w}$ be a preimage of${\displaystyle v}$, then in${\displaystyle \{v,w\}}$ basis,, and so the flow is ashearing along the${\displaystyle v}$ direction. In this case,${\displaystyle \mathrm{tr}A=0}$.
  • If${\displaystyle \ker A\ne \mathrm{im}A}$, then let${\displaystyle v}$ span${\displaystyle \ker A}$ and let${\displaystyle w}$ span${\displaystyle \mathrm{im}A}$, then in${\displaystyle \{v,w\}}$ basis, for some nonzero real number${\displaystyle a}$.
    • If${\displaystyle \mathrm{tr}A>0}$, then it is unstable, diverging at a rate of${\displaystyle a}$ from${\displaystyle \ker A}$ along parallel translates of${\displaystyle \mathrm{im}A}$.
    • If, then it is stable, converging at a rate of${\displaystyle a}$ to${\displaystyle \ker A}$ along parallel translates of${\displaystyle \mathrm{im}A}$.

If${\displaystyle detA\ne 0}$, we first find theJordan normal form of the matrix, to obtain a basis${\displaystyle \{v,w\}}$ in which${\displaystyle A}$ is one of three possible forms:

• where${\displaystyle a,b\ne 0}$.
  • If${\displaystyle a,b>0}$, then${\displaystyle \{\begin{array}{l}4detA-(\mathrm{tr}A{)}^2=-(a-b{)}^2\le 0\\ detA=ab>0\end{array}}$. The origin is asource, with integral curves of form${\displaystyle y=c{x}^{b/a}}$
  • Similarly for. The origin is asink.
  • If${\displaystyle a>0>b}$ or, then, and the origin is asaddle point. with integral curves of form${\displaystyle y=c{x}^{-|b/a|}}$.
• where${\displaystyle a\ne 0}$. This can be further simplified by a change-of-basis with, after which. We can explicitly solve for${\displaystyle \dot{X}=AX}$ with. The solution is${\displaystyle X(t)=e^{At}X(0)}$ with. This case is called the "degenerate node". The integral curves in this basis are central dilations of${\displaystyle x=y\ln y}$, plus the x-axis.
  • If${\displaystyle \mathrm{tr}A>0}$, then the origin is andegenerate source. Otherwise it is adegenerate sink.
  • In both cases,${\displaystyle 4detA-(\mathrm{tr}A{)}^2=0}$
• where${\displaystyle a>0,\theta \in (-\pi ,\pi ]}$. In this case,${\displaystyle 4detA-(\mathrm{tr}A{)}^2=(2a\sin \theta {)}^2\ge 0}$.
  • If${\displaystyle \theta \in (-\pi ,-\pi /2)\cup (\pi /2,\pi ]}$, then this is aspiral sink. In this case,. The integral lines arelogarithmic spirals.
  • If${\displaystyle \theta \in (-\pi /2,\pi /2)}$, then this is aspiral source. In this case,${\displaystyle \{\begin{array}{l}4detA-(\mathrm{tr}A{)}^2>0\\ \mathrm{tr}A>0\end{array}}$. The integral lines arelogarithmic spirals.
  • If${\displaystyle \theta =-\pi /2,\pi /2}$, then this is arotation ("neutral stability") at a rate of${\displaystyle a}$, moving neither towards nor away from origin. In this case,${\displaystyle \mathrm{tr}A=0}$. The integral lines are circles.

The summary is shown in the stability diagram on the right. In each case, except the case of${\displaystyle 4detA-(\mathrm{tr}A{)}^2=0}$, the values${\displaystyle (\mathrm{tr}A,detA)}$ allows unique classification of the type of flow.

For the special case of${\displaystyle 4detA-(\mathrm{tr}A{)}^2=0}$, there are two cases that cannot be distinguished by${\displaystyle (\mathrm{tr}A,detA)}$. In both cases,${\displaystyle A}$ has only one eigenvalue, withalgebraic multiplicity 2.

• If the eigenvalue has a two-dimensional eigenspace (geometric multiplicity 2), then the system is acentral node (sometimes called a "star", or "dicritical node") which is either a source (when${\displaystyle \mathrm{tr}A>0}$) or a sink (when).^[2]
• If it has a one-dimensional eigenspace (geometric multiplicity 1), then the system is adegenerate node (if${\displaystyle detA>0}$) or ashearing flow (if${\displaystyle detA=0}$).

When${\displaystyle \mathrm{tr}A=0}$, we have${\displaystyle det{e}^{At}=e^{\mathrm{tr}(A)t}=1}$, so the flow is area-preserving. In this case, the type of flow is classified by${\displaystyle detA}$.

• If${\displaystyle detA>0}$, then it is a rotation ("neutral stability") around the origin.
• If${\displaystyle detA=0}$, then it is a shearing flow.
• If, then the origin is a saddle point.

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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, smalloscillations as in the case of apendulum. In a system withdamping, 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 itslinearization.

Letf:R →R be acontinuously differentiable function with a fixed pointa,f(a) =a. Consider the dynamical system obtained by iterating the functionf:

$${\displaystyle x_{n+1}=f(x_n),n=0,1,2,\dots .}$$

The fixed pointa is stable if theabsolute value of thederivative off ata is strictly less than 1, and unstable if it is strictly greater than 1. This is because near the pointa, the functionf has alinear approximation with slopef'(a):

$${\displaystyle f(x)\approx f(a)+f^{\prime }(a)\left(x-a\right).}$$

Thus

$${\displaystyle \Rightarrow f^{\prime }(a)\approx \frac{x_{n+1}-a}{x_n-a}}$$

which means that the derivative measures the rate at which the successive iterates approach the fixed pointa or diverge from it. If the derivative ata is exactly 1 or −1, then more information is needed in order to decide stability.

There is an analogous criterion for a continuously differentiable mapf:Rⁿ →Rⁿ with a fixed pointa, expressed in terms of itsJacobian matrix ata,J_a(f). If alleigenvalues ofJ are real or complex numbers with absolute value strictly less than 1 thena is a stable fixed point; if at least one of them has absolute value strictly greater than 1 thena is unstable. Just as forn=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 fordiffeomorphisms of asmooth manifold.

The stability of fixed points of a system of constant coefficientlinear differential equations of first order can be analyzed using theeigenvalues of the corresponding matrix.

Anautonomous system

$${\displaystyle x^{\prime }=Ax,}$$

wherex(t) ∈Rⁿ andA is ann×n matrix with real entries, has a constant solution

$${\displaystyle x(t)=0.}$$

(In a different language, the origin0 ∈Rⁿ is an equilibrium point of the corresponding dynamical system.) This solution is asymptotically stable ast → ∞ ("in the future") if and only if for all eigenvaluesλ ofA,Re(λ) < 0. Similarly, it is asymptotically stable ast → −∞ ("in the past") if and only if for all eigenvaluesλ ofA,Re(λ) > 0. If there exists an eigenvalueλ ofA withRe(λ) > 0 then the solution is unstable fort → ∞.

The stability of a linear system can be determined by solving the differential equation to find the eigenvalues, or without solving the equation by using theRouth–Hurwitz stability criterion. The eigenvalues of a matrix are the roots of itscharacteristic polynomial. A polynomial in one variable with real coefficients is called aHurwitz polynomial if the real parts of all roots are strictly negative. TheRouth–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 theHartman–Grobman theorem.

Suppose thatv is aC¹-vector field inRⁿ which vanishes at a pointp,v(p) = 0. Then the corresponding autonomous system

$${\displaystyle x^{\prime }=v(x)}$$

has a constant solution

$${\displaystyle x(t)=p.}$$

LetJ_p(v) be then×nJacobian matrix of the vector fieldv at the pointp. If all eigenvalues ofJ have strictly negative real part then the solution is asymptotically stable. This condition can be tested using theRouth–Hurwitz criterion.

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A general way to establishLyapunov stability or asymptotic stability of a dynamical system is by means ofLyapunov functions.

• Chaos theory – Field of mathematics and science based on non-linear systems and initial conditions
• Ship stability – Ship response to disturbance from an upright condition
• Lyapunov stability – Property of a dynamical system where solutions near an equilibrium point remain so
• Hyperstability
• Linear stability – State of linear equations
• Orbital stability – Solution to a partial differential equation which remains close to the initial data
• Stability criterion
• Stability radius – Concept in mathematics
• Structural stability – Concept in mathematics
• von Neumann stability analysis – Numerical analysis procedure

• Philip Holmes and Eric T. Shea-Brown (ed.)."Stability".Scholarpedia.

• Stable Equilibria by Michael Schreiber,The Wolfram Demonstrations Project.

• Stability theory
• Limit sets
• Mathematical and quantitative methods (economics)

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