Equilibrium

Revision as of 00:39, 21 April 2007

Figure 1: Illustration of a stable and unstable equilibrium point.

Equilibrium (or equilibrium point) of a dynamical system is a solution that does not change with time. For example, each motionless pendulum in Fig.<ref>eq_pend</ref> is at equilibrium. Geometrically, equilibria are points in the system's phase space. Equilibria could be stable or unstable, as e.g., in Fig.<ref>eq_pend</ref>.

More precisely, the dynamical system \[ x'=f(x) \] has an equilibrium solution \(x(t)=x_{\rm e}\) if \(f(x_{\rm e})=0\). If the system is one-dimensional, i.e., \(x\) is a scalar, then equilibria are just zeros of the function \(f(x)\). If it is \(n\)-dimensional system, i.e., \[x=(x_1,\ldots,x_n)\] and \[f(x)=(f_1(x_1,\ldots,x_n),\ldots,f_n(x_1,\ldots,x_n))\], then equilibria must satisfy \[f_i(x_1,\ldots,x_n)=0\] for all \(i\).

Equilibria are sometimes called fixed points. Most mathematicians refer to equilibria as time-independent solutions of continuous dynamical systems, e.g., ODEs, and to fixed points as time-independent solutions of iterated maps \(x(t+1) = f(x(t))\).

Jacobian Matrix

Stability of most equilibria are determined by the sign of eigenvalues of the Jacobian matrix. Jacobian Matrix of a system of ODEs at an equilibrium \(x=x_{\rm e}\) is the matrix of partial derivatives of the right-hand side with respect to state variables

<math Jacobian>

J=D_xf = \left(\frac{\partial f_i}{\partial x_j}\right) = \begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \ldots & \frac{\partial f_1}{\partial x_n} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \ldots & \frac{\partial f_2}{\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial f_n}{\partial x_1} & \frac{\partial f_n}{\partial x_2} & \ldots & \frac{\partial f_n}{\partial x_n} \end{pmatrix} </math> where all derivatives are evaluated at the equilibrium. Its eigenvalues determine linear stability properties of the equilibrium.

An equilibrium is stable if all eigenvalues have negative real parts; It is unstable if at least one eigenvalue has positive real part.

Hyperbolic Equilibria

The equilibrium is said to be hyperbolic if all eigenvalues of the Jacobian matrix have non-zero real parts. The Hartman-Grobman Theorem says that dynamics of \[ x'=f(x) \] in a local neighborhood of a hyperbolic equilibrium is topologically conjugate (equivalent) to the dynamics of the linear system \[ y' = Jy \], where \(J\) is the Jacobian matrix at the equilibrium.

If at least one eigenvalue of the Jacobian matrix is zero or has zero real part, then the equilibrium is said to be non-hyperbolic. Such equilibria arise when the system undergoes a bifurcation. In practice, one often has to consider non-hyperbolic equilibria with all eigenvalues having negative or zero real parts. These equilibria are sometimes referred to as being critical. Stability of critical equilibria cannot be determined from the signs of the eigenvalues of the Jacobian matrix; it depends on the nonlinear terms of \(f\).

Types of Equilibria

Figure 2: Equilibria of a one-dimensional system \(x'=f(x)\) are the points where \(f(x)=0\).

One-Dimensional Space

Consider a one-dimensional (scalar) dynamical system \[x'=f(x)\], \(x\in\R^1\) with a continuous function \(f(x)\). Its equilibria are the zeros of the function \(f(x)\), as illustrated in Fig.<ref>equil1</ref>. The Jacobian matrix at each equilibrium is \(J=f'(x)\). An equilibrium is stable when \(f'(x)<0\); that is, the slope of \(f\) is negative. It is unstable when \(f'(x)>0\). The left two equilibria in the figure are hyperbolic (\(f'(x) \neq 0\)), the others are non-hyperbolic because the slope (eigenvalue) is zero. Nevertheless, a non-hyperbolic equilibrium of a one-dimensional system is stable if the function changes the sign from positive to negative at the equilibrium.

Two-Dimensional Space

Consider a two-dimensional (planar) system \[x_1' = f_1(x_1, x_2)\] \[x_2' = f_2(x_1, x_2)\]. The Jacobian matrix has the form \[ J= \begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} \end{pmatrix} \].

Figure 3: Classification of equilibria of a two-dimensional dynamical system according to the trace (\(\tau\)) and the determinant (\(\Delta\)) of the Jacobian matrix. The shaded region corresponds to stable equilibria. (modified from Izhikevich 2007).

It has two eigenvalues, which are either both real or complex-conjugate. A hyperbolic equilibrium can be

• Node when both eigenvalues are real and of the same sign; The node is stable when the eigenvalues are negative and unstable when they are positive;
• Saddle when eigenvalues are real and of opposite signs; The saddle is always unstable;
• Focus (sometimes called spiral) when eigenvalues are complex-conjugate; The focus is stable when the eigenvalues have negative real part and unstable when they have positive real part.

Let \[ \tau = {\rm tr } J = \frac{\partial f_1}{\partial x_1}+ \frac{\partial f_2}{\partial x_2} \] be the trace and \[ \Delta = {\rm det } J = \frac{\partial f_1}{\partial x_1}\frac{\partial f_2}{\partial x_2} - \frac{\partial f_1}{\partial x_2}\frac{\partial f_2}{\partial x_1} \] be the determinant of the Jacobian matrix. Figure <ref>summary2d</ref> summarizes the types of equilibria. The axes \(\tau=0\) and \(\Delta=0\) correspond to nonhyperbolic equilibria that arise during Andronov-Hopf and Saddle-Node Bifurcation, respectively.

Figure 4: Examples of equilibria in \(\R^3\).

Three-Dimensional Space

The Jacobian matrix of a three-dimensional system has 3 eigenvalues, one of which must be real and the other two can be either both real or both complex-conjugate. Depending on the types and signs of the eigenvalues, there are a few interesting cases illustrated in Fig.<ref>equil3d</ref>. A hyperbolic equilibrium can be

• Node when all eigenvalues are real and have the same sign; The node is stable (unstable) when the eigenvalues are negative (positive);
• Saddle when all eigenvalues are real and at least one of them is positive and at least one is negative; Saddles are always unstable.
• Focus-Node when it has one real eigenvalue and a pair of complex-conjugate eigenvalues, and all eigenvalues have real parts of the same sign; The equilibrium is stable (unstable) when the sign is negative (positive).
• Saddle-Focus when it has one real eigenvalue with the sign opposite to the sign of the real part of a pair of complex-conjugate eigenvalues; This type of equilibrium is always unstable.

Higher-Dimensional Spaces

All the cases described above could also occur in higher-dimensional systems. For example, 2 real and 2 complex-conjugate eigenvalues of the same sign would result in a focus-node equilibrium, whereas of the opposite signs would result in saddle-focus equilibrium. In addition, there could be focus-focus equilibria, which correspond to two pairs of complex-conjugate eigenvalues, saddle-focus-focus, etc.

Nonhyperbolic Equilibria