# Stability of equilibria

Post-publication activity

Curator: Emmanuil E. Shnol

This article considers systems of ordinary differential equations of the form $\tag{1} x'=F(x)\ , x\in\R^n\ ,$

having an equilibrium $$x=c$$ whose stability cannot be determined by the linearized system $$y'=Ay\ ,$$ $$A=F'(c)\ .$$

## Equilibria

Consider a system of ordinary differential equations of the form (1) having a time-independent solution $$x(t)=c\ .$$ The trajectory of such a solution consists of one point, namely $$c\ ,$$ and such a point is called an equilibrium. Equilibria can be stable or unstable. Stable equilibria have practical meaning since they correspond to the existence of a certain observable regime. This needs to be defined mathematically, and we will use the definition provided by Lyapunov. The definition is

• local, i.e., in some neighborhood of the equilibrium $$c\ ;$$
• it describes only asymptotic behavior of solutions, i.e., when $$t\rightarrow+\infty\ ;$$
• it contains two notions: neutral stability (Lyapunov stability) and asymptotic stability;
• it takes into account only perturbations of the initial conditions of the system (1).

One could say that the Lyapunov definition considers only instantaneous or pulsed perturbations of the system.

Our intuitive notion of stability of equilibrium is more general than the one defined by Lyapunov: Such an equilibrium should persist under the influences of small external perturbations ("noise"), and not only small perturbations of the initial condition. It is remarkable that asymptotically stable equilibria are also stable in this more general sense – under small persistent perturbations (Massera 1949; Malkin 1958).

An asymptotically stable equilibrium is the simplest example of an attractor of (1).

## Linearization

Since stability is defined in a local neighborhood of the equilibrium, we can linearize system (1) near $$c$$ to obtain $\tag{2} y'=Ay \ .$

The asymptotic dynamics of the linearized system (2) depends on the eigenvalues $$\lambda$$ of the Jacobian matrix $$A=F'(c)\ .$$ When all eigenvalues have non-zero real parts, the equilibrium is called hyperbolic, and non-hyperbolic if at least one eigenvalue has zero real part.

• If all eigenvalues $$\lambda$$ have negative real parts, then all solutions of (2) approach zero exponentially. It seems clear that taking into account nonlinear terms in $$y$$ (in some neighborhood of equilibrium $$c$$) would not change the asymptotic behavior of the solutions, so the origin $$y=0$$ of the linear system, and hence the equilibrium $$c$$ of the nonlinear system (1) is attracting.
• If at least one eigenvalue has a positive real part, then the majority of solutions of the linearized system grow exponentially and it seems clear that the linearized equilibrium $$y=0\ ,$$ and hence the equilibrium $$c$$ of the nonlinear system (1) is unstable.

Both propositions are valid but their proof is not simple. It was first given by A.M. Lyapunov in 1892 (see Lyapunov functions).

If all eigenvalues have non-positive real parts, but there is $$\lambda$$ with zero real part, then one needs to consider non-linear terms of the Taylor series of $$F(x)\ .$$ Simple examples show that nonlinear terms could guarantee the asymptotic stability absent in the linear case, or could cancel the weak instability of the linear system that occurs when the Jacobian matrix has eigenvalues with algebraic multiplicity on the imaginary axis. Hence, the stability of equilibria in such critical cases is the simplest problem of nonlinear dynamics. Below we refer to a non-hyperbolic equilibrium whose stability is determined by the nonlinear terms of $$F(x)$$ as a critical equilibrium.

## Reduced Normal Forms

Intuitively, it is clear that there are certain variables corresponding to the zero and purely imaginary eigenvalues that determine the stability of a critical equilibrium. The Center Manifold Theorem allows one to reduce the full system to a smaller subsystem containing the Jacobian matrix with all eigenvalues on the imaginary axis (zero or purely imaginary). In addition, the system can be reduced to a normal form up to a certain order . Finally, the normal form can be reduced further by removing terms that do not affect the stability of the equilibrium. For brevity, we refer below to resulting polynomial systems as normal forms.

## Two Major Critical Cases

Two major critical cases correspond to one zero eigenvalue and a pair of purely imaginary eigenvalues of the Jacobian matrix.

• If $$\lambda=0\ ,$$ then the normal form is one-dimensional$y'=a_2y^2\ .$ The equilibrium $$y=0$$ (corresponding to the equilibrium $$x=c$$) is unstable for any $$a_2\neq \ 0\ .$$ If $$a_2=0\ ,$$ then the normal form is $$y'=a_3y^3\ .$$ Apparently, the equilibrium $$y=0$$ is stable when $$a_3<0$$ and unstable when $$a_3>0\ .$$
• If $$\lambda=\pm i \omega\ ,$$ the normal form is two-dimensional, conveniently written in the complex form

$\tag{3} z'=i \omega z + Qz|z|^2\ ,$

where $$z=y_1+iy_2\ .$$ Notice that (3) does not have quadratic and most of the cubic terms. The equilibrium $$z=0$$ is asymptotically stable when $$q=\mbox{Re}\,Q<0$$ and unstable when $$q>0\ .$$

We do not consider here the method of finding the coefficients $$a$$ and $$Q$$ from the right-hand side of the original system (see Saddle-Node Bifurcation and Andronov-Hopf Bifurcation). The lessons from these two simplest cases have a general meaning:

1. The existence of zero eigenvalues of the Jacobian matrix $$A=F'(c)$$ leads to the instability of the equilibrium "with probability 1". Stability is possible only when additional conditions in the form $$\Phi({\mathbf a})=0$$ are imposed. Here, $${\mathbf a}$$ is the set of Taylor coefficients of $$F(x)$$ at the point $$x=c$$ up to a certain order, which may not be known a priori, and $$\Phi$$ is a polynomial in some of these coefficients. In the simplest case of one $$\lambda=0\ ,$$ the condition is $$a_2=0\ .$$
2. If all critical $$\lambda$$ are purely imaginary, then asymptotic stability is possible "with positive probability": if a few conditions of the form $$\Phi({\mathbf a})<0$$ are satisfied. The condition $$\Phi({\mathbf a})=0$$ restricts the class of systems and can lead to instability; see below.

## More Complicated Example

Suppose the Jacobian matrix $$A=F'(c)$$ has two pairs of purely imaginary eigenvalues $$\lambda_{1,2}=\pm i \omega_1\ ,$$ $$\lambda_{3,4}=\pm i \omega_2\ ,$$ and all other eigenvalues (if they are) have negative real parts. Let $$\omega_2>\omega_1>0\ ,$$ $$\omega_2 \neq 2\omega_1\ ,$$ and $$\omega_2 \neq 3\omega_1\ .$$ Then, the normal form is $z_1=i \omega_1z_1+z_1(Q_{11}|z_1|^2+Q_{12}|z_2|^2) \ ,$ $z_2=i \omega_2z_2+z_2(Q_{21}|z_1|^2+Q_{22}|z_2|^2) \ .$ Let $$q_{kl}=\mbox{Re}\,Q_{kl}\ .$$ The equilibrium $$z=0$$ (corresponding to the equilibrium $$x=c$$ in system (1)) is asymptotically stable if all the following conditions are satisfied:

1. $$q_{11}<0\ ,$$
2. $$q_{22}<0\ ,$$
3. If $$q_{12}$$ and $$q_{21}$$ are positive, then $$q_{12}q_{21}-q_{11}q_{22}<0\ .$$

If at least one of $$q_{12}$$ or $$q_{21}$$ is not positive, then conditions 1 and 2 suffice.

The equilibrium $$z=0$$ is unstable if any of the conditions above contain the sign $$>$$ instead of the sign $$<\ .$$ This criterion does not tell anything in the case when at least one inequality above is replaced by an equality (i.e., has $$=$$ sign). In particular, if there is a resonant relation $$\omega_2=2\omega_1\ ,$$ then the normal form has quadratic terms and the critical equilibrium is unstable "with probability 1", that is, under the condition $$\Phi\neq0\ .$$

## Classification of Critical Cases

Each critical case corresponds to a certain number, $$k\ ,$$ of equality conditions ($$\Phi=0$$) imposed on the Taylor coefficients of $$F(x)$$ at $$x=c\ .$$ The number $$k$$ is called the co-dimension. Cases of co-dimension $$k$$ appear naturally when one considers families of dynamical systems $$x'=F(x, s)$$ with $$s=(s_1,\ldots,s_k)$$ (Arnold 1983).

• Co-dimension 1 corresponds to the two simplest and the most important cases (Section Two Major Critical Cases): one zero eigenvalue and a pair of pure complex eigenvalues.
• Co-dimension 2 includes the example in Section More Complicated Example.
• Co-dimension 3 includes the equilibrium in Section More Complicated Example, if, in addition, one of the following conditions is satisfied:
• The resonance condition $$\omega_2=2\omega_1\ ,$$ or
• The condition $$q_{11}=0\ .$$

Notice that $$q_{11}$$ is a quite complicated function of linear, quadratic, and cubic Taylor coefficients of $$F(x)\ .$$ That is, the conditions are imposed not only on the Jacobian matrix, but also on the nonlinear part of the dynamical system.

All cases of co-dimension $$k\leq3$$ are known for the general system (1). Arnold and Ilyashenko (1988) explain why $$k=3$$ is a natural upper bound of the general theory. See also Khazin and Shnol (1991).

Systems with critical equilibria are not structurally stable: When parameters are perturbed, phase portraits of such systems undergo qualitative change - a bifurcation - near the equilibria. The first critical case ($$\lambda=0$$) corresponds to saddle-node bifurcation; the second case ($$\lambda = \pm i \omega$$) corresponds to Andronov-Hopf bifurcation; critical cases of co-dimensions 2 and 3 correspond to more complicated bifurcations, some of which are not well-understood.

## Special Classes of Systems

### Hamiltonian Systems

Classification according to co-dimension is quite different in Hamiltonian systems: The existence of any number of purely imaginary eigenvalues ($$\lambda = \pm i \omega$$) of the Jacobian matrix still corresponds to $$k=0\ .$$ Cases of co-dimension $$k>0\ ,$$ corresponding to the existence of resonant relations among the frequencies $$\omega_j\ ,$$ must be considered separately. Moreover, one needs to ask different kind of questions about dynamics of Hamiltonian systems, since asymptotic stability of equilibria in such systems is impossible. However, the instability due to resonances may be interesting.

### Systems with Symmetry

Suppose system (1) admits a finite (or compact) symmetry group $$G\ ,$$ consisting of linear transformations$F(gx)=gF(x)$ for all $$g\in G\ .$$ Suppose the equilibrium $$x=c$$ is invariant with respect to $$G\ ,$$ i.e., $$gc=c\ .$$ If the symmetry group $$G$$ is non-Abelian, then the Jacobian matrix $$A$$ typically has eigenvalues with algebraic multiplicity $$n_\star>1\ .$$ Possible values of $$n_\star$$ depend on complex irreducible representations of $$G$$ corresponding to the action of $$G$$ on $$\R^n\ .$$ Hence, co-dimension 1 critical equilibria in one-parameter families of systems with symmetry can have multiple zero or purely imaginary eigenvalues. The difficulty of determining the stability of such equilibria depends mostly on the value of $$n_\star\ .$$ All cases of $$n_\star=2$$ and all cases of $$\lambda=0$$ with multiplicity 3 are known (Shnol 1999). There are many cases of $$n_\star=3$$ with $$\lambda=\pm i \omega$$ and many cases of $$n_\star=4$$ with $$\lambda=0\ .$$ The complete understanding of dynamics in all these cases is highly unlikely, but some of them may be very interesting.