Stability theory

Classification of stability types

If ${\displaystyle \det A=0}$, 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 ${\displaystyle \operatorname {im} A}$ are both one-dimensional.
  • If ${\displaystyle \ker A=\operatorname {im} A}$, then let ${\displaystyle v}$ span ${\displaystyle \ker A}$, and let ${\displaystyle w}$ be a preimage of ${\displaystyle v}$, then in ${\displaystyle \{v,w\}}$ basis, ${\displaystyle A={\begin{bmatrix}0&1\\0&0\end{bmatrix}}}$, and so the flow is a shearing along the ${\displaystyle v}$ direction. In this case, ${\displaystyle \operatorname {tr} A=0}$.
  • If ${\displaystyle \ker A\neq \operatorname {im} A}$, then let ${\displaystyle v}$ span ${\displaystyle \ker A}$ and let ${\displaystyle w}$ span ${\displaystyle \operatorname {im} A}$, then in ${\displaystyle \{v,w\}}$ basis, ${\displaystyle A={\begin{bmatrix}0&0\\0&a\end{bmatrix}}}$ for some nonzero real number ${\displaystyle a}$.
    • If ${\displaystyle \operatorname {tr} A>0}$, then it is unstable, diverging at a rate of ${\displaystyle a}$ from ${\displaystyle \ker A}$ along parallel translates of ${\displaystyle \operatorname {im} A}$.
    • If ${\displaystyle \operatorname {tr} A<0}$, then it is stable, converging at a rate of ${\displaystyle a}$ to ${\displaystyle \ker A}$ along parallel translates of ${\displaystyle \operatorname {im} A}$.

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

• ${\displaystyle {\begin{bmatrix}a&0\\0&b\end{bmatrix}}}$ where ${\displaystyle a,b\neq 0}$.
  • If ${\displaystyle a,b>0}$, then ${\displaystyle {\begin{cases}4\det A-(\operatorname {tr} A)^{2}=-(a-b)^{2}\leq 0\\\det A=ab>0\end{cases}}}$. The origin is a source, with integral curves of form ${\displaystyle y=cx^{b/a}}$
  • Similarly for ${\displaystyle a,b<0}$. The origin is a sink.
  • If ${\displaystyle a>0>b}$ or ${\displaystyle a<0<b}$, then ${\displaystyle \det A<0}$, and the origin is a saddle point. with integral curves of form ${\displaystyle y=cx^{-|b/a|}}$.
• ${\displaystyle {\begin{bmatrix}a&1\\0&a\end{bmatrix}}}$ where ${\displaystyle a\neq 0}$. This can be further simplified by a change-of-basis with ${\displaystyle C={\begin{bmatrix}1/a&0\\0&1\end{bmatrix}}}$, after which ${\displaystyle A=a{\begin{bmatrix}1&1\\0&1\end{bmatrix}}}$. We can explicitly solve for ${\displaystyle {\dot {X}}=AX}$ with ${\displaystyle A=a{\begin{bmatrix}1&1\\0&1\end{bmatrix}}}$. The solution is ${\displaystyle X(t)=e^{At}X(0)}$ with ${\displaystyle e^{At}=e^{at}{\begin{bmatrix}1&at\\0&1\end{bmatrix}}}$. 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 ${\displaystyle \operatorname {tr} A>0}$, then the origin is an degenerate source. Otherwise it is a degenerate sink.
  • In both cases, ${\displaystyle 4\det A-(\operatorname {tr} A)^{2}=0}$
• ${\displaystyle a{\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}}}$ where ${\displaystyle a>0,\theta \in (-\pi ,\pi ]}$. In this case, ${\displaystyle 4\det A-(\operatorname {tr} A)^{2}=(2a\sin \theta )^{2}\geq 0}$.
  • If ${\displaystyle \theta \in (-\pi ,-\pi /2)\cup (\pi /2,\pi ]}$, then this is a spiral sink. In this case, ${\displaystyle {\begin{cases}4\det A-(\operatorname {tr} A)^{2}>0\\\operatorname {tr} A<0\end{cases}}}$. The integral lines are logarithmic spirals.
  • If ${\displaystyle \theta \in (-\pi /2,\pi /2)}$, then this is a spiral source. In this case, ${\displaystyle {\begin{cases}4\det A-(\operatorname {tr} A)^{2}>0\\\operatorname {tr} A>0\end{cases}}}$. The integral lines are logarithmic spirals.
  • If ${\displaystyle \theta =-\pi /2,\pi /2}$, then this is a rotation ("neutral stability") at a rate of ${\displaystyle a}$, moving neither towards nor away from origin. In this case, ${\displaystyle \operatorname {tr} A=0}$. The integral lines are circles.

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

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

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

Area-preserving flow

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

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

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:

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

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):

${\displaystyle f(x)\approx f(a)+f'(a)(x-a).}$

Thus

${\displaystyle {\begin{aligned}x_{n+1}-a&=f(x_{n})-a\\&\approx f(a)+f'(a)(x_{n}-a)-a\\&=a+f'(a)(x_{n}-a)-a\end{aligned}}}$

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

which means that the derivative measures the rate at which the successive iterates approach the fixed 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, J_a(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

${\displaystyle x'=Ax,}$

where x(t) ∈ Rⁿ and A is an n×n matrix with real entries, has a constant solution

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

(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

${\displaystyle x'=v(x)}$

has a constant solution

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

Let J_p(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.