Overdamped pendulum

Equation (1) is a second-order system, but in the overdamped limit of extremely large b, it may be approximated by a first-order system (see Section 3.5 and Exercise 4.4.1), In this limit the inertia term mlrQ is negligible and so (1) becomes 

To think about this problem physically, you should imagine that the pendulum is immersed in molasses. The torque F enables the pendulum to plow through its vis- 

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Condensed-matter physics (Josephson junction, charge-density waves) Mechanics (Overdamped pendulum driven by a constant torque)... 

We now consider a simple mechanical example of a nonuniform oscillator an overdamped pendulum driven by a constant torque. Let 0 denote the angle between the pendulum and the downward vertical, and suppose that d increases counterclockwise (Figure 4.4.1). 

Torsional spring) Suppose that our overdamped pendulum is connected to a torsional spring. As the pendulum rotates, the spring winds up and generates... 

Although quantum mechanics is required to explain the origin of the Josephson effect, we can nevertheless describe the dynamics of Josephson junctions in classical terms. Josephson junctions have been particularly useful for experimental studies of nonlinear dynamics, because the equation governing a single junction is the same as that for a pendulum In this section we will study the dynamics of a single junction in the overdamped limit. In later sections we will discuss underdamped junctions, as well as arrays of enormous numbers of junctions coupled together. 

This section deals with a physical problem in which both homoclinic and infinite-period bifurcations arise. The problem was introduced back in Sections 4.4 and 4.6. At that time we were studying the dynamics of a damped pendulum driven by a constant torque, or equivalently, its high-tech analog, a superconducting Josephson junction driven by a constant current. Because we weren t ready for two-dimensional systems, we reduced both problems to vector fields on the circle by looking at the heavily overdamped limit of negligible mass (for the pendulum) or negligible capacitance (for the Josephson junction). 

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