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 (5.20)–(5.21). 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

If \(\Delta \lt 0\text{,}\) the eigenvalues of the Jacobian are complex. Since the real part of (5.22) is negative, these equilibrium solutions are spiral sinks.

If \(\Delta \gt 0\text{,}\) we have two distinct real eigenvalues. Since

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 (Figure 5.27).

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

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 (Figure 5.27).

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

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.

are \(\pm i\text{,}\) the linearization has a center at the origin. The phase plane consists of circles about the origin (Figure 5.28). Notice that the linearization does not depend on \(\alpha\text{.}\)

Now let us consider what happens to system (5.24)–(5.25) if we consider different values of \(\alpha\text{.}\) If \(\alpha = 5\text{,}\) the situation is quite different that the linearization of our system. A solution curve spirals out from the origin as \(t \to \infty\) (Figure 5.29). 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 (Figure 5.30).

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,

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 \(t \to -\infty\text{,}\) equation (5.26) tells us that \(r(t) \to 0\text{.}\) Thus, any solution to the system (5.24)–(5.25) 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, \(V\) 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

\begin{equation*}
\dot{V} = \nabla V \cdot {\mathbf F},
\end{equation*}

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.^{ 21 }See Figures 1 and 2 in John Polking, Albert Boggess, and David Arnold. {\it Differential Equations}. Prentice Hall, Upper Saddle River, NJ, 2001, p. 611. 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 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.^{ 22 }The argument that we have made here also works in higher dimensions.

Now suppose that \(V\) is a real-valued 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

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 a asymptotically stable equilibrium solution or a sink.

If \(\alpha \lt 0\text{,}\) then \(\dot{V}\) is negative definite on \({\mathbb R}^2\text{.}\) Theorem 5.31 tells us that the origin is a stable equilibrium point.

The function \(V\) in Theorem 5.31 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.

Thus, \(S\) increases at point on the solution curve where the gradient of \(S\) is nonzero. That is, \(S\) increases at every point on the solution curves except at the equilibrium points.

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 real-valued 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 \(S\text{.}\)

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 a 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.

\(S\) 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.

The 60-story, 790-foot mirror-glass John Hancock Tower, New England's tallest building, was completed in 1976 and was designed by Henry N. Cobb, who founded the firm Pei Cobb Freed & Partners with famed architect I.M. Pei. (Figure 5.32).^{ 23 }The John Hancock Tower is officially known by its street address, 200 Clarendon. The building suffered many problems during construction, the most notorious of which was how the original glass windows were attached. In early 1972 when construction was still underway, one of the 500-pound windows popped out of the building and committed suicide on the sidewalk below. In all more than 100 of the building's windows suffered the same fate. Fortunately, no one was injured. Initially, the architects and engineers thought thought that the problem was caused by the building's tendancy to sway excessively in high winds. However, they later determined that the falling-window problem was caused by the air space between the double-paned windows and pressure differentials between the interior and exterior of the building. The problem was solved by replacing all of the windows with single sheets of fully tempered glass. During the repairs, the windows were replaced with plywood, and the building was nicknamed the “Plywood Palace.”

Another flaw in the design of the building resulted in an extreme amount of swaying. In fact, office workers in the upper stories of the building complained of motion sickness. The building actually twisted as it was swaying back and forth. Engineers determined that the building's natural sway period was dangerously close to the period of its torsion. To remedy than problem, the engineers installed a pair of tuned mass dampeners on the 58th floor of the building. This is the floor of concrete that you see near the top of the structure (Figure 5.32). In addition, 1,500 tons of steel braces were installed to keep the building from falling over in a high wind.

To construct a simple model of a tuned-mass damper, we will consider a mass-spring system connected to a fixed wall. The mass is allowed to slide freely with no damping. We can think of this mass as the top of the building moving back and forth. To this first mass, we attach a second mass by using a spring and a dashpot. This second mass is called the tuned mass-damper. If we “tune” the mass-damper by choosing the appropriate spring, we can remove energy from the system and the motion of the first mass (the building) will decrease.

We will first model our system without a dashpot. If we let \(L_1\) and \(L_2\) be the natural lengths of the springs, the first spring will exert a force \(-k_1(x_1 - L_1)\text{.}\) The second spring pushes the two masses in opposite directions according to \(\pm k_2(x_2 - x_1 - L_2)\text{.}\) Thus, we can use Newton's second law to derive two equations

We can show that a solution to the system, \((x_1(t), x_2(t), p_1(t), p_2(t))\text{,}\) is constant on \(H\text{.}\)^{ 24 }In general, a Hamiltonian system is a system of \(2n\) equations of the form

for \(i = 1, 2, \ldots, n\text{,}\) where \(H(x_1, \ldots, x_n, y_1, \ldots y_n\) is a real-valued differentiable function on \({\mathbb R}^{2n}\) such that \(H\) is nonconstant on every open ball in \({\mathbb R}^{2n}\text{.}\) We can show that \(H\) is constant on solution curves of the system.

Now suppose that we add a damping term to our system,

\begin{equation*}
\pm b \left( \frac{dx_2}{dt} - \frac{dx_1}{dt} \right) = \pm b \left( \frac{p_2}{m_2} - \frac{p_1}{m_1} \right).
\end{equation*}

\begin{equation}
\frac{dH}{dt} = - b \left( \frac{p_2}{m_2} - \frac{p_1}{m_1} \right)^2 = - b \left( \frac{dx_2}{dt} - \frac{dx_1}{dt} \right)^2.\label{equation-nonlinear03-tuned-mass-damper-3}\tag{5.29}
\end{equation}

Equation (5.29) tells us that \(dH/dt \leq 0\text{.}\) Moreover, \(dH/dt \lt 0\) whenever the distance between the two masses is changing. Thus, energy decreases whenever the second mass is moving relative to the first. Thus, if the wind or an earthquake starts our building (\(m_1\)) swaying back and forth, then the tuned mass-damper (\(m_2\)) located on one of the top floors of the building will start to move relative to the building and energy will be removed from the system by the dampers.

Of course, \(m_1\) and \(k_1\) involve the building and are set by the architects and engineers. We, however, are free to choose \(m_2\text{,}\) \(k_2\) and \(b\text{.}\) We want to choose \(b\) fairly large so that there is a rapid loss of energy; i.e., the magnitude of \(dH/dt\) is large. We should choose \(m_2\) large enough so that this mass oscillates with respect to \(m_1\text{.}\) If we choose \(m_2\) two small, then the strong damper will almost serve as a rigid connection between \(m_1\) and \(m_2\text{.}\) Therefore, we wish to choose \(m_2\) to be large so that we guaranteed that this mass will oscillate with respect to \(m_1\text{.}\) We should, however, remember that \(m_2\) is sitting on top of a very tall building. Finally, we should choose \(k_2\) to maximize the rate at which the second mass oscillates with respect to the first and to maximize the oscillations of \(m_2\text{.}\) If we choose \(k_2\) so that the second spring is in resonance with the first, the oscillations of the first mass force the second at its resonance frequency. Thus, we will have relatively large oscillations of the second mass and a large loss of energy.

Let us consider example. Suppose that \(m_1 = 1\text{,}\) \(k_1 = 1\text{,}\) We will also choose \(m_2\) to be 0.05 and \(b = 0.1\text{.}\) If the initial conditions for our system are

we will choose \(k_2\) to tune our system. To do this we will plot the time \(t\) that it takes for the amplitude of the oscillations of \(m_1\) to reach and stay below 2.5 for various values of \(k_2\text{.}\)^{ 25 }See Figure 5.42 in Paul Blanchard, Robert L. Devaney, and Glen R. Hall. Differential Equations, Second edition. Brooks/Cole, Pacific Grove, CA, 2002, p.~499. If we choose \(k_2 \approx 0.05\text{,}\) then we can minimize the time to be \(t \approx 120\text{.}\)