195bpend.htm

Ecology / Mathematics 195b.

From the Pendulum to the Lynx & the Hare

VIII. Case Studies.

Physics:

From the Pendulum ...

A. We study the dynamics of an ideal pendulum. Assume the following:

1. Bob a point mass, m.

2. Weightless string of length, l.

3. Angle of displacement from the vertical, q, measured in radians.

4. No friction.

5. No external forces.

B. Recall that the arc length, s, is given by

s = l q

Example: Half the circumferance of a circle of radius, l, is p l.

C. Enumerate the forces & derive a second order differential equation describing motion of the bob.

1. F is the tension in the string.

2. mg is the weight of the bob. This force (mass x acceleration is a force) can be resolved into radial (along the string) and azimuthal (along s) components.

a. F_r = m g cos q

b. F_a = m g sin q

3. Since force equals mass x acceleration,

F_a = - mg sin q = (d²s/dt²)

where (d²s/dt²) is the second derivative of s with respect to t, i.e., the rate at which the bob accelerates. In symbols,

(d²s/dt²) = (d/dt) v = (d/dt)(ds/dt)

Thus,

(d²s/dt²) = - g sin q

But since s = l q, we may write

(d²q/dt²) = - (g/l) sin q

4. In elementary physics, you learn the following:

a. For small initial displacements, the period of oscillation is (l/g)^1/2.

b. With increasing displacement, the period increases.

D. Non-dimensionalization. It is useful to rewrite our equation in simpler form by imposing a transformation of variables which gets rid of the units. This is a common first step in the mathematical analysis of dynamical systems.

1. In this regard, we note that radians, our measure of angle are dimensionless - i.e., it just wouldn't do for the circumferance of a circle, C = 2 p r, to have units of centimeters-radians!

2. If we measure length in cm and time in sec, both sides thus have units sec^-2.

3. Define "dimensionless time," t = t(g/l)^1/2.

* * *

Optional Homework Exercise: (5 points). Show that this allows us to rewrite our equation as

(d²q/dt²) = - sin q

* * *

E. Rewrite as a pair of 1st order differential equations.

1. Our equation is a second order differential equation (DE), i.e., it has a second derivative.

2. It is generally true that we can rewrite an n^th order DE as n first order equations, i.e., equations which only have first derivatives.

3. In the present case, we define the angular velocity, v = dq/dt. Then we have

dq/dt = v
dv/dt = - sin q

4. This is a well-defined dynamical system in the sense that we have been using the term: both state variables occur on both sides of the equality.

F. Phase Portrait

1. We now plot the motion of the pendulum, for representative initial velocities and displacements, in the q-v plane. For convenience, we consider only those values of q on the interval, [-p,p].

a. The point (0,0) thus corresponds to the pendulum at rest with the bob pointing down.

b. The points (p,0) and (-p,0) also correspond to the pendulum at rest, but in this case, the bob points up.

2. Imagine an initial, small displacement, q(0), from the vertical. We release the pendulum, imparting no velocity, i.e., v(0) = 0, and it begins oscillating.

a. We distinguish right to left from left to right motion by attaching a minus or plus sign to the velocity.

b. As the pendulum oscillates, it traces out a closed loop in the theta-v plane.

3. Now imagine a large initial dispacement, with q(0) = p - e (e small) and v(0)=0. The pendulum still goes back and forth, but now, in the phase plane, the loops go almost to +p and -p

4. Finally, imagine an initial displacement with the bob pointing straight up, i.e., q(0) = +p or -p and a non-zero initial velocity, i.e., v(0) = +v₀ or v(0) = -v₀. In this case, the pendulum revolves forever. In terms of the phase plane portrait, this accounts for the non-repeating orbits near the top and the bottom.

* * *

Optional Homework Exercise: (5 points for each of three). Now imagine that the pendulum is subject to friction and what is called "driving." We suppose that friction depends on velocity and enters into the equations as - bv and that there is a wind which acts on the pendulum with constant force, - a. Thus the second order equation of motion is

(d²q/dt²) = - sin q - a - b v

a. Sketch the phase portrait when the coefficient, b, of friction is non- zero. b. Sketch the phase portrait when both the coefficients of friction, b, and driving, a, are non-zero. (Hint: There are two possibilities in this case.)

* * * G. Effects of periodic forcing.

1. Modify the force equation as follows:

(d²/dt²) q = - sin q + F(T)

where F(T) is a periodic function. Specifically, choose

F(T) = e cos(2 p wT).

a. e measures the amplitude of forcing; w measures the frequency.

b. For concreteness, imagine that the forcing results from a wind, the velocity of which varies periodically.

2. We study the consequences of this addition to the model by constructing "stroboscopic maps" wherein we sample solution curves at intervals equal to the period,

P = w^-1,

of the forcing.

H. By way of prelude we construct such a map for the special case, e = 0.

1. As one moves vertically from the rest point, (0,0), the period of the pendulum increases from 0 to infinity.

2. We restrict our attention to the periodic orbits (pendulum oscillates). If we sample at intervals equal to the period of forcing to be imposed when e > 0, the resulting constructions will either be

a. Closed loops, or

b. Cycles composed of finite numbers of points.

3. Which possibility obtains depends on whether or not the period of the orbit in question is rationally related to the period of forcing.

4. Think of the orbits as "degenerate," i.e., infinitely thin, tori. By computing a stroboscopic map, we have divided the tori into two categories:

a. Rational tori (the cycles)

b. Irrational tori (the closed loops).

5. Question: Why would we do this?

6. Answer: Because it allows us to comprehend the consequences of perturbing e away from zero.

I. Nonintegrability.

1. In the absence of forcing, the pendulum corresponds to what is called an integrable dynamical system.

2. The orbits of n-dimensional integrable systems are said to be "regular," i.e., cycles or tori of dimension n/2.

3. Upon perturbing e away from 0, we induce a transition to what is called non-integrability.

4. The principal consequences (see accompanying figure) are as follows:

a. Irrational tori persist, at least initially.

b. Rational tori are replaced by complex structures called "island chains."

1. Each chain is organized about a pair of periodic orbits, the rotation number of which equals that of the torus the chain replaces.

2. One of these cycles is neutrally stable; the other is a saddle.

3. Surrounding the saddle cycle are secondary tori and additional (secondary) island chains. The secondary chains are surrounded by tertiary chains, etc., with the consequence that the island chains are self-similar.

4. In the vicinity of the saddle cycles are chaotic orbits. For small values of e, these orbits are confined to thin layers, so-called "stochastic regions" by the surviving (primary) irrational tori.

5. With further increases in e, we observe the following:

a. The irrational tori begin breaking down.

b. The chaotic orbits wander widely and merge to form larger regions of chaos called the "stochastic sea."

c. The island chains also begin breaking down.

6. In short, the transition from integrable to non-integrable dynamics induces infinite dynamical complexity where previously none existed. For an animated view of one such orbit, click on the figure at the right.

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