Section 5.3 More Nonlinear Mechanics
Objectives
To understand that the equation for the nonlinear pendulum with damping
\begin{align*} \frac{d \theta}{dt} & = v\\ \frac{dv}{dt} & =  \frac{b}{m} v  \frac{g}{L} \sin \theta. \end{align*}can be analyzed by examining \(dH/dt\text{,}\) where \(H\) is the Hamiltonian function for the ideal pendulum. The function \(H\) for this system is an example of a Lyapunov function.To understand how to use Lyapunov functions to determine the stability of an equilibrium solution for a nonlinear system.
The equation for the nonlinear pendulum with damping is
As a system, we can model the pendulum as
This system is not Hamiltonian since
but
We shall have to invent new techniques to analyze such systems.
Subsection 5.3.1 The Nonlinear Pendulum and Damping
To find the nullclines of the nonlinear pendulum, let
The \(\theta\)nullcline is \(v = 0\text{,}\) while the \(v\)nullcline is
It follows that the equilibrium solutions for the nonlinear pendulum occur at
This makes sense since the pendulum should not move if the bob is initially hanging downward (\(\theta = 2 \pi n\)) or is at the very top or the very bottom of a swing (\(\theta = (2n + 1)\pi\)). Since our first goal is to determine the nature of each equilibrium solution, we will compute the Jacobian of the system (5.3.1)â€“(5.3.2). This is just
At the equilibrium solutions \((\theta, v) = (0,0), (\pm 2\pi, 0), (\pm 4\pi, 0), \ldots\text{,}\) the pendulum is hanging downward, and the Jacobian matrix becomes
On the other hand, if \((\theta, v) = (\pm \pi, 0), (\pm 3\pi, 0), (\pm 5\pi, 0), \ldots\text{,}\) the pendulum is at the top of its swing, and the Jacobian matrix is
We will first examine the case where the pendulum is hanging downward. The characteristic polynomial of \(J_1\) is
Thus, the eigenvalues of \(J_1\) are
We can analyze the nature of these eigenvalues by examining the sign of
If \(\Delta \lt 0\text{,}\) the eigenvalues of the Jacobian are complex. Since the real part of (5.3.3) is negative, these equilibrium solutions are spiral sinks.
If \(\Delta \gt 0\text{,}\) we have two distinct real eigenvalues. Since
\begin{equation*} \left( \frac{b}{2m} \right)^2  \frac{g}{L} \lt \left( \frac{b}{2m} \right)^2, \end{equation*}we know that\begin{equation*} \left( \frac{b}{2m} \right) \gt \sqrt{\left( \frac{b}{2m} \right)^2  \frac{g}{L} }. \end{equation*}Thus, both of our eigenvalues must be negative. Therefore, we have a nodal sink.If \(\Delta = 0\text{,}\) we have a single real negative eigenvalue. Thus, we also have a sink.
Consequently, if we assume that \(b\) is small, then \(\Delta \lt 0\) and we will only have spiral sinks (FigureÂ 5.3.1).
Now let us consider the type of equilibrium solutions that we will obtain when the pendulum is standing upright. These solutions will occur at \((\theta, v) = (\pm \pi, 0), (\pm 3\pi, 0), (\pm 5\pi, 0), \ldots\text{.}\) The characteristic polynomial of the Jacobian matrix \(J_2\) at these points is
hence, the eigenvalues of \(J_2\) are
Furthermore, we have distinct real eigenvalues since
In fact, the eigenvalues will have opposite sign
Thus, the equilibrium solutions are saddles.
Subsection 5.3.2 Lyapunov Functions
The function
is a Hamiltonian function for the ideal pendulum
since \(d H/ dt = 0\text{.}\) However, if \(b \gt 0\text{,}\) then our system
has damping and
Thus, \(H\) is decreasing whenever \(v \neq 0\text{.}\) Hence, solution curves in the \(\theta v\)plane cross the level sets of \(H\) moving from larger to smaller \(H\) values.
We can now devise a strategy for sketching the phase plane of the damped pendulum. If \(b/m\) and \(v\) are both small, the value of \(H\) decreases slowly along the solutions (FigureÂ 5.3.1).
The function \(H\) in the case of the damped pendulum is an example of a Lyapunov function. Specifically, a function \(L(x, y)\) is called a Lyapunov function for the system
if for every solution of the system, \((x(t), y(t))\text{,}\) that is not an equilibrium solution of the system,
with strict inequality except possible for a discrete set of \(t\)s.
As an example, let us return to the damped harmonic oscillator
If \(p = 0\text{,}\) then
is a Hamiltonian function for our system. Recall that we also call \(H\) the energy function of the system. However, if \(p \gt 0\) and \((y(t), v(t))\) is a solution for our system, we have
Consequently, \(H(y(t), v(t))\) decreases at a nonzero rate (except when \(v = 0\)), and \(H\) is a Lyapunov function. The level sets of \(H\) are ellipses in the \(yv\)plane. As \(H\) decreases, the energy dissipates and the ellipses become spiral sinks.
It is easy to see that the system
has an equilibrium solution at the origin no matter what the value of \(\alpha\) is. The Jacobian of this system is
Since our equilibrium solution is the origin,
and the linearization of our system at the origin is
Since the eigenvalues of
are \(\pm i\text{,}\) the linearization has a center at the origin. The phase plane consists of circles about the origin (FigureÂ 5.3.2). Notice that the linearization does not depend on \(\alpha\text{.}\)
Now let us consider what happens to systemÂ (5.3.5)â€“(5.3.6) if we consider different values of \(\alpha\text{.}\) If \(\alpha = 5\text{,}\) the situation is quite different than the linearization of our system. A solution curve spirals out from the origin as \(t \to \infty\) (FigureÂ 5.3.3). As \(t \to  \infty\text{,}\) the solution curve spirals back into the origin, but it seems to stop before actually reaching the origin. If \(\alpha = 5\) on the other hand, we seem to have the opposite behavior with the solution curves spiraling into the origin as \(t \to \infty\text{.}\) As before, the solutions do not seem to reach the origin (FigureÂ 5.3.4).
Suppose that \((x(t), y(t))\) is a solution to the nonlinear system. The function
is the distance of a point on the solution curve to the origin in the \(xy\)plane. To see how \(r\) changes as \(t \to \pm \infty\text{,}\) we can compute the derivative of \(r\text{.}\) Actually, it is easier to work with the equation \(r(t)^2 = x(t)^2 + y(t)^2\text{.}\) Thus,
Since \((r^2)' = 2r r'\text{,}\) we have
for \(r \neq 0\text{.}\)
EquationÂ (5.3.7) is separable, and it is quite easy to determine the solution as
However, we do not need to know this solution to determine the nature of the equilibrium solution at the origin. If \(\alpha \gt 0\) and \(t \to \infty\text{,}\) equationÂ (5.3.7) tells us that \(r(t) \to 0\text{.}\) Thus, any solution to the systemÂ (5.3.5)â€“(5.3.6) we have a spiral sink at the origin if \(\alpha = 5\text{.}\) Even though linearization fails to tell us the nature of the equilibrium solution at the origin, we were able to determine the nature of the equilibrium solution with further analysis.
We will now try to exploit what we have learned from our last example and from Hamiltonian systems to see if it is possible to analyze more general systems. If we consider solutions, \((x(t), y(t))\text{,}\) of the system
we might ask how a function \(V(x, y)\) varies along the solution curve. We already have an answer if our system is Hamiltonian, and \(V\) is the corresponding Hamiltonian function. In this case \(dV/dt = 0\text{.}\) In general, we know that
Thus, if we let
we know that
Thus, \(V\) is increasing along a solution curve if \(\dot{V}(x, y) \gt 0\) and decreasing along a solution curve if \(\dot{V}(x, y) \lt 0\text{.}\) Our example suggests that we can determine this information without finding the solution.
Recall that the gradient of a function \(V: {\mathbb R}^2 \rightarrow {\mathbb R}\) is
If
then
is the directional derivative of \(V\) in the direction of \({\mathbf F}\text{.}\)
Let us use this new information about \(V\) to obtain information about equilibrium solutions of our system. We do know that \(V(x, y)\) graphs as a surface in \({\mathbb R}^3\) and
gives the contour lines or level curves of the surface in the \(xy\)plane.^{â€‰1â€‰} We also know that the gradient of \(V\) points in the direction that \(V\) is increasing the fastest and that the gradient is orthogonal to the level curves of \(V\text{.}\) Thus, if \(\dot{V}(x, y) \gt 0\text{,}\) we know that \(V\) is increasing in the direction of the vector field \({\mathbf F}\) and the elevation of the solution curve through \((x, y)\) in \({\mathbb R}^3\) is increasing. That is, the solution curve is traveling uphill. Similarly, if \(\dot{V}(x, y) \lt 0\text{,}\) we know that the solution curve at \((x, y)\) is going downhill.^{â€‰2â€‰}
Now suppose that \(V\) is a realvalued function defined on a set \(S\) in the \(xy\)plane, where the point \({\mathbf x}_0 = (x_0, y_0)\) is in \(S\text{.}\) We say that \(V\) is positive definite if \(V({\mathbf x}) \gt 0\) for all \({\mathbf x}\) in \(S\text{,}\) where \({\mathbf x} \neq {\mathbf x}_0\text{,}\) and \(V\) is positive semidefinite if \(V({\mathbf x}) \geq 0\text{.}\) Similarly, we say that \(V\) is negative definite and negative semidefinite if \(V({\mathbf x}) \lt 0\) and \(V({\mathbf x}) \leq 0\text{,}\) respectively.
For example, if we consider the system for a harmonic oscillator
then the energy function,
is positive definite.
Theorem 5.3.5.
Suppose that the system
has an equilibrium solution at \((x_0, y_0)\text{.}\) Let \(V\) be a continuously differentiable function defined on a neighborhood \(U\) of \((x_0, y_0)\) that is positive definite with minimum at \((x_0, y_0)\text{.}\)
If \(\dot{V}\) is negative semidefinite on \(U\text{,}\) then \((x_0, y_0)\) is a stable equilibrium solution. That is, any solution that starts near the equilibrium solution will stay near the equilibrium solution.
If \(\dot{V}\) is negative definite on \(U\text{,}\) then \((x_0, y_0)\) is an asymptotically stable equilibrium solution or a sink.
Recall our example,
The function \(V(x, y) = x^2 + y^2\) is positive definite on \({\mathbb R}^2\text{,}\) with an isolated minimum at the origin. We can compute
If \(\alpha \lt 0\text{,}\) then \(\dot{V}\) is negative definite on \({\mathbb R}^2\text{.}\) TheoremÂ 5.3.5 tells us that the origin is a stable equilibrium point.
The function \(V\) in TheoremÂ 5.3.5 is called a Lyapunov function. If we compare Theorem 1 to using linearization to determine stability of an equilibrium solution, we will find that we can apply this result where linearization fails. Also, Lyapunov functions are defined on a domain \(U\text{,}\) where linearization only tells us what happens on a small neighborhood around the equilibrium solution. Unfortunately, there are no general ways of finding Lyapunov functions.
Subsection 5.3.3 Gradient Systems
Let \(S\) be a realvalued function on the \(xy\)plane. The gradient of \(S\) is
The system
is a gradient system if
For example, the system
is a gradient system, where
Now, let us see what happens on the solution curves of this gradient system. If \((x(t), y(t))\) is a solution curve,
Thus, \(S\) increases at the point on the solution curve where the gradient of \(S\) is nonzero. That is, \(S\) increases at every point on the solution curve except at the equilibrium points.
In general, suppose that
is a gradient system. Since
we know that \(S\) increases on every solution to the system except at the critical points of \(S(x(t), y(t))\text{.}\)
Let us see what this means in terms of the linearization of the system. The Jacobian matrix of \((f, g)\) is
Since \(S_{xy} = S_{yx}\text{,}\) the matrix \(J\) must have the form
where \(\alpha = S_{xx}\text{,}\) \(\beta = S_{xy}\text{,}\) and \(\gamma = S_{yy}\text{.}\) The characteristic polynomial of \(J\) is
Therefore, the eigenvalues are
Since we have real eigenvalues, a gradient system can have no spiral sources, spiral sinks, or centers.
For example, \(x^2 + y^2\) is a Lyapunov function for the system
However, the origin is a spiral sink, so this system cannot be a gradient system.
Subsection 5.3.4 Important Lessons
The equation for the nonlinear pendulum with damping
\begin{align*} \frac{d \theta}{dt} & = v\\ \frac{dv}{dt} & =  \frac{b}{m} v  \frac{g}{L} \sin \theta. \end{align*}can be analyzed by examining \(dH/dt\text{,}\) where \(H\) is the Hamiltonian function for the ideal pendulum. The function \(H\) for this system is an example of a Lyapunov function.
Let \(V\) be a realvalued function defined on a set \(S\) in the \(xy\)plane such that the point \({\mathbf x}_0 = (x_0, y_0)\) is in \(S\) and \(V({\mathbf x}_0) = 0\text{.}\)
We say that \(V\) is positive definite if \(V({\mathbf x}) \gt 0\) for all \({\mathbf x}\) in \(S\text{,}\) where \({\mathbf x} \neq {\mathbf x}_0\text{.}\)
We say that \(V\) is positive semidefinite if \(V({\mathbf x}) \geq 0\) for all \({\mathbf x}\) in \(S\text{.}\)
We say that \(V\) is negative definite if \(V({\mathbf x}) \lt 0\) for all \({\mathbf x}\) in \(S\text{,}\) where \({\mathbf x} \neq {\mathbf x}_0\text{.}\)
We say that \(V\) is negative semidefinite if \(V({\mathbf x}) \leq 0\) for all \({\mathbf x}\) in \(S\text{.}\)

Suppose that the system
\begin{align*} \frac{dx}{dt} & = f(x, y)\\ \frac{dy}{dt} & = g(x, y) \end{align*}has an equilibrium solution at \((x_0, y_0)\text{.}\) Let \(V\) be a continuously differentiable function defined on a neighborhood \(U\) of \((x_0, y_0)\) that is positive definite with minimum at \((x_0, y_0)\text{.}\)
If \(\dot{V}\) is negative semidefinite on \(U\text{,}\) then \((x_0, y_0)\) is a stable equilibrium solution. That is, any solution that starts near the equilibrium solution will stay near the equilibrium solution.
If \(\dot{V}\) is negative definite on \(U\text{,}\) then \((x_0, y_0)\) is an asymptotically stable equilibrium solution or a sink.
We can use these results to analyze the behavior of equilibrium solutions where linearization fails. The function \(V\) is called a Lyapunov function. We have no general methods for finding Lyapunov functions.
The system
\begin{align*} \frac{dx}{dt} & = f(x, y)\\ \frac{dy}{dt} & = g(x, y) \end{align*}is a gradient system if\begin{align*} f(x, y) & = \frac{\partial S}{\partial x}\\ g(x, y) & = \frac{\partial S}{\partial y}, \end{align*}where \(S\) is a realvalued function on the \(xy\)plane. Since\begin{equation*} \frac{dS}{dt} (x(t), y(t)) = \left( \frac{\partial S}{\partial x} \right)^2 + \left( \frac{\partial S}{\partial y} \right)^2 \geq 0, \end{equation*}\(S\) increases on every solution to the system except at the critical points of \(S(x(t), y(t))\text{.}\) Since the eigenvalues of a gradient system are real, a gradient system has no spiral sources, spiral sinks, or centers.
Reading Questions 5.3.5 Reading Questions
1.
Explain in your own words what a Lyapunov function is.
2.
Expalin in your own words what a gradient system is.
Exercises 5.3.6 Exercises
1.
This is an exercise.