3.6: The Stability of Fixed Points in Nonlinear Systems

We are now interested in studying the stability of the equilibrium solutions of the nonlinear pendulum. Along the way we will develop some basic methods for studying the stability of equilibria in nonlinear systems.

We begin with the linear differential equation for damped oscillations as given earlier in Equation (3.9). In this case, we have a second order equation of the form

\[x'' + bx' + w^2x. \nonumber \]

Using the methods of Chapter 2, this second order equation can be written as a system of two first order equations:

\[\begin{aligned}
x' &= y \\
y' &= -by - w^2x.
\end{aligned} \label{3.20} \]

This system has only one equilibrium solution, \(x = 0, y = 0\).

Turning to the damped nonlinear pendulum, we have the system

\[\begin{aligned}
x' &= y \\
y' &= -by - w^2 \sin x.
\end{aligned} \label{3.21} \]

This system also has the equilibrium solution, \(x=0, y=0\). However, there are actually an infinite number of solutions. The equilibria are determined from \(y=0\) and \(-b y-\omega^{2} \sin x=0\). This implies that \(\sin x=0\). There are an infinite number of solutions: \(x=n \pi, n=0, \pm 1, \pm 2, \ldots\) So, we have an infinite number of equilibria, \((n \pi, 0), n=0, \pm 1, \pm 2, \ldots\)

Next, we need to determine their stability. To do this we need a more general theory for nonlinear systems. We begin with the \(n-\) dimensional system

\[\mathbf{x}^{\prime}=\mathbf{f}(\mathbf{x}), \quad \mathbf{x} \in \mathrm{R}^{n} \label{3.22} \]

Here \(\mathbf{f}: \mathrm{R}^{n} \rightarrow \mathrm{R}^{n}\). We define fixed points, or equilibrium solutions, of this system as points \(\mathbf{x}^{*}\) satisfying

\(\mathbf{f}\left(\mathbf{x}^{*}\right)=\mathbf{0}\)

The stability in the neighborhood of fixed points can now be determined. We are interested in what happens to solutions of our system with initial conditions starting near a fixed point. We can represent a point near a fixed point in the form \(\mathbf{x}=\mathbf{x}^{*}+\boldsymbol{\xi}\), where the length of \(\boldsymbol{\xi}\) gives an indication of how close we are to the fixed point. So, we consider that initially, \(|\boldsymbol{\xi}| \ll 1\).

As the system evolves, \(\boldsymbol{\xi}\) will change. The change of \(\boldsymbol{\xi}\) in time is in turn governed by a system of equations. We can approximate this evolution as follows. First, we note that

\[\mathbf{x}^{\prime}=\boldsymbol{\xi}^{\prime} \nonumber \]

Next, we have that

\[\mathbf{f}(\mathbf{x})=\mathbf{f}\left(\mathbf{x}^{*}+\boldsymbol{\xi}\right) \nonumber \]

We can expand the right side about the fixed point using a multidimensional version of Taylor's Theorem. Thus, we have that

\[\mathbf{f}\left(\mathbf{x}^{*}+\boldsymbol{\xi}\right)=\mathbf{f}\left(\mathbf{x}^{*}\right)+D \mathbf{f}\left(\mathbf{x}^{*}\right) \boldsymbol{\xi}+O\left(|\boldsymbol{\xi}|^{2}\right) \nonumber \]

Here \(D \mathbf{f}\) is the Jacobian matrix, defined as

\(D \mathbf{f}=\left(\begin{array}{cccc}
\dfrac{\partial f_{1}}{\partial x_{1}} & \dfrac{\partial f_{1}}{\partial x_{2}} & \cdots & \dfrac{\partial f_{1}}{\partial x_{n}} \\
\dfrac{\partial f_{2}}{\partial x_{1}} & \ddots & \ddots & \vdots \\
\vdots & \ddots & \ddots & \vdots \\
\dfrac{\partial f_{n}}{\partial x_{1}} & \cdots & \cdots & \dfrac{\partial f_{n}}{\partial x_{n}}
\end{array}\right)\)

Noting that \(\mathbf{f}\left(\mathbf{x}^{*}\right)=\mathbf{0}\), we then have that system (3.22) becomes

\[\xi^{\prime} \approx D \mathbf{f}\left(\mathbf{x}^{*}\right) \boldsymbol{\xi} \label{3.23} \]

It is this equation which describes the behavior of the system near the fixed point. We say that system (3.22) has been linearized or that Equation (3.23) is the linearization of system (3.22).

Example 3.4. As an example of the application of this linearization, we look at the system

\[\begin{aligned}
&x^{\prime}=-2 x-3 x y \\
&y^{\prime}=3 y-y^{2}
\end{aligned} \label{3.24} \]

We first determine the fixed points:

\[\begin{aligned}
&0=-2 x-3 x y=-x(2+3 y) \\
&0=3 y-y^{2}=y(3-y)
\end{aligned} \label{3.25} \]

From the second equation, we have that either \(y=0\) or \(y=3\). The first equation then gives \(x=0\) in either case. So, there are two fixed points: \((0,0)\) and \((0,3)\).

Next, we linearize about each fixed point separately. First, we write down the Jacobian matrix.

\[D \mathbf{f}(x, y)=\left(\begin{array}{cc}
-2-3 y & -3 x \\
0 & 3-2 y
\end{array}\right) \label{3.26} \]

1. Case I \((0,0)\).

In this case we find that

\[D \mathbf{f}(0,0)=\left(\begin{array}{cc}
-2 & 0 \\
0 & 3
\end{array}\right) . \label{3.27} \]

Therefore, the linearized equation becomes

\[\xi^{\prime}=\left(\begin{array}{cc}
-2 & 0 \\
0 & 3
\end{array}\right) \boldsymbol{\xi} . \label{3.28} \]

This is equivalently written out as the system

\[\begin{aligned}
&\xi_{1}^{\prime}=-2 \xi_{1} \\
&\xi_{2}^{\prime}=3 \xi_{2}
\end{aligned} \label{3.29} \]

This is the linearized system about the origin. Note the similarity with the original system. We emphasize that the linearized equations are constant coefficient equations and we can use earlier matrix methods to determine the nature of the equilibrium point. The eigenvalues of the system are obviously \(\lambda=-2,3\). Therefore, we have that the origin is a saddle point.

2. Case II \((0,3)\)

In this case we proceed as before. We write down the Jacobian matrix and look at its eigenvalues to determine the type of fixed point. So, we have that the Jacobian matrix is

\[D \mathbf{f}(0,3)=\left(\begin{array}{cc}
-2 & 0 \\
0 & -3
\end{array}\right) \text {. } \label{3.30} \]

Here, we have the eigenvalues \(\lambda=-2,-3\). So, this fixed point is a stable node.

This analysis has given us a saddle and a stable node. We know what the behavior is like near each fixed point, but we have to resort to other means to say anything about the behavior far from these points. The phase portrait for this system is given in Figure 3.14. You should be able to find the saddle point and the node. Notice how solutions behave in regions far from these points.

Figure 3.14. Phase plane for the system \(x^{\prime}=-2 x-3 x y, y^{\prime}=3 y-y^{2}\).

We can expect to be able to perform a linearization under general conditions. These are given in the Hartman-Großman Theorem:

Theorem 3.5.

A continuous map exists between the linear and nonlinear systems when \(D \mathbf{f}\left(\mathbf{x}^{*}\right)\) does not have any eigenvalues with zero real part.

Generally, there are several types of behavior that one can see in nonlinear systems. One can see sinks or sources, hyperbolic (saddle) points, elliptic points (centers) or foci. We have defined some of these for planar systems. In general, if at least two eigenvalues have real parts with opposite signs, then the fixed point is a hyperbolic point. If the real part of a nonzero eigenvalue is zero, then we have a center, or elliptic point.

Example 3.6. Return to the Nonlinear Pendulum

We are now ready to establish the behavior of the fixed points of the damped nonlinear pendulum in Equation (3.21). The system was

\[\begin{aligned}
x^{\prime} &=y \\
y^{\prime} &=-b y-\omega^{2} \sin x .
\end{aligned} \label{3.31} \]

We found that there are an infinite number of fixed points at \((n \pi, 0), n= 0, \pm 1, \pm 2, \ldots\)
We note that the Jacobian matrix is

\[D \mathbf{f}(x, y)=\left(\begin{array}{cc}
0 & 1 \\
-\omega^{2} \cos x & -b
\end{array}\right) . \label{3.32} \]

Evaluating this at the fixed points, we find that

\[D \mathbf{f}(n \pi, 0)=\left(\begin{array}{cc}
0 & 1 \\
-\omega^{2} \cos n \pi & -b
\end{array}\right)=\left(\begin{array}{cc}
0 & 1 \\
\omega^{2}(-1)^{n+1} & -b
\end{array}\right) \label{3.33} \]

There are two cases to consider: \(n\) even and \(n\) odd. For the first case, we find the eigenvalue equation

\[\lambda^{2}+b \lambda+\omega^{2}=0 \nonumber \]

This has the roots

\[\lambda=\dfrac{-b \pm \sqrt{b^{2}-4 \omega^{2}}}{2} . \nonumber \]

For \(b^{2}<4 \omega^{2}\), we have two complex conjugate roots with a negative real part. Thus, we have stable foci for even $n$ values. If there is no damping, then we obtain centers.
In the second case, \(n\) odd, we have that

\[\lambda^{2}+b \lambda-\omega^{2}=0 . \nonumber \]

In this case we find

\[\lambda=\dfrac{-b \pm \sqrt{b^{2}+4 \omega^{2}}}{2} . \nonumber \]

Since \(b^{2}+4 \omega^{2}>b^{2}\), these roots will be real with opposite signs. Thus, we have hyperbolic points, or saddles.

In Figure (3.15) we show the phase plane for the undamped nonlinear pendulum. We see that we have a mixture of centers and saddles. There are orbits for which there is periodic motion. At \(\theta=\pi\) the behavior is unstable. This is because it is difficult to keep the mass vertical. This would be appropriate if we were to replace the string by a massless rod. There are also unbounded orbits, going through all of the angles. These correspond to the mass spinning around the pivot in one direction forever. We have indicated in the figure solution curves with the initial conditions \(\left(x_{0}, y_{0}\right)=(0,3),(0,2),(0,1),(5,1)\).
When there is damping, we see that we can have a variety of other behaviors as seen in Figure (3.16). In particular, energy loss leads to the mass settling around one of the stable fixed points. This leads to an understanding as to why there are an infinite number of equilibria, even though physically the mass traces out a bound set of Cartesian points. We have indicated in the Figure (3.16) solution curves with the initial conditions \(\left(x_{0}, y_{0}\right)=(0,3),(0,2),(0,1),(5,1)\).

Figure 3.15. Phase plane for the undamped nonlinear pendulum. Solution curves are shown for initial conditions \((x_0, y_0) = (0, 3), (0, 2), (0, 1), (5, 1)\).