Conserved quantities
1. The Predator-Prey system
The Predator-Prey system models the relations between two populations where the second population feeds on the first. In the simplest model, the prey has a fixed growth rate that is reduced by the population of the predator:
\dot{x} = (a - b y)x,
and the predator has a negative growth rate (poplulation goes down in the absence of food) that is increased by the presence of the prey:
\dot{y} = (-c + ex)y,
where a, b, c, e \ge 0 are positive parameters.
Calculations show that the linearized system at the fixed point (c/e, a/b) is an ellipstic centre. This is a degenerate type, therefore we cannot decide conclusively the phase portrait of the non-linear system. Some other techniques are needed.
On the part of the phase space where \dot{x} \ne 0, we can divide the two equations to get
\frac{\dot{y}}{\dot{x}} = \frac{y(-c + e x)}{x(a - by)}.
We can separate the variables
\left( \frac{a}{y} - b\right) \dot{y} = \left( -\frac{c}{x} + e\right) \dot{x}
then integrate both sides to get (we consider only positive values for x, y)
a \log(y(t)) - b y(t) = - c \log(x(t)) + ex(t) + C.
This means the function
K(x, y) = a \log y + b y + c \log x - e x
is a constant along trajectories x(t), y(t). To plot the phase portrait, we note that all the trajectories must travel along the level curves of the function K.
Figure 1. Phase portrait of the predator prey system
The idea of using conserved quantities to study dynamics is very common.
Definition 1.1.
Let \dot{\bx} = \bfF(\bx), \bx \in \R^n be an ODE. A function G: \R^n \to \R is a conserved quantity for the system if for every solution \bx(t) of the system, G(\bx(t)) = \const, or equivalently
\frac{d}{dt} G(\bx(t)) = 0.
Remark 1.2.
A conserved quantity can sometimes be found using separation of variables as in our example, or sometimes it comes from physical considerations (conservation of mass, energy, momentum etc).
It is also important to note that conserved quantities do not necessarily exist for all systems. There are systems for which there are no (smooth) conserved quantities.
2. Classical mechanical systems
For a particle in one-dimensional space moving under laws of Newtonian mechanics, the position x(t) of the particle satisfies
m \ddot{x} = f,
where m is the mass of the particle, and f is the force exerted on x at a given time. If the force exerted on the particle depends on the position of the particle (called a force field), we can write f as a function of x, from which we get the second order ODE
m \ddot{x} - f(x) = 0.\qquad (2.1)
Set y = \dot{x}, we get the 2-dimensional system
\dot{x} = y, \quad \dot{y} = \frac{1}{m} f(x).
A conserved quantity can be found using separation of variables. Since
\frac{\dot{x}}{\dot{y}} = \frac{my}{f(x)},
we get
m y \dot{y} - f(x) \dot{x} = 0.
Let V(x) be an anti-derivative of - f(x), integrating both sides leads to
\frac12 m y^2 + V(x) = C.
The function
E(x, y) = \frac12 m y^2 + V(x)
is the energy function of the system, where \frac12 m y^2 is called the kinetic energy and V(x) the potential energy. The fact that E(x, y) is conserved is a manifestation of the physical principle conservation of energy. Since V'(x) = - f(x), equation (2.1) is often written in the following form
m \ddot{x} + V'(x) = 0.
Equations of this type are called classical mechanical systems or sometimes classical systems.
Example 2.1.
The Harmonic Oscillator
m \ddot{x} + k x = 0
has the potential function V(x) = \frac{k}{2} x^2. The energy is
\frac12 m y^2 + V(x) = \frac12 m y^2 + \frac12 kx^2.
For the Harmonic oscillator, all trajectories moves along the ellipses m y^2 + k x^2 = C.
Example 2.2.
The mathematical pendulum:
\ddot{x} + \frac{g}{l} \sin x = 0.
The potential is V(x) = - \frac{g}{l} \cos(x).
We will come back to this example after exploring the general method for phase portraits of classical systems.
2.1. Level curves of the energy function
Remark 2.3.
Since classical systems are second order equations, there are two dimensions involved. One is the dimension of the variable x, which we will call spatial dimension but in physics is more commonly called the degrees of freedom. The dimension of the phase space is always twice the spatial dimension.
All our examples so far has spatial dimension 1 and phase space of dimension 2.
To analyze the level curves of the energy
\frac12 m y^2 + V(x) = C,
we rewrite the equation as
y = \pm \sqrt{\frac{2}{m}(C - V(x))}.
We can sketch these curves by looking at the positive part of the function C - V(x).
Consider the set \{x \st C - V(x) > 0\}, which is a union of open intervals (an open set in \R is always a union of open intervals). Let (a, b) be one of such intervals, the necessarily V(a) = V(b) = C (otherwise the interval should have been larger). On the interval (a, b),
\sqrt{\frac{2}{m} (C - V(x))}, \quad
- \sqrt{\frac{2}{m} (C - V(x))}
are both positive smooth functions. However, the way that the two graphs connect at the points (a, 0) and (b, 0) are different.- If V'(a) \ne 0, then V'(a) < 0 (since V(x) < C in (a, b)). We have\frac{d}{dx} \sqrt{\frac{2}{m} (C - V(x))} = \frac{-V'(x)}{m\sqrt{\frac{2}{m} (C - V(x))}} \to - \infty, \quad x \to a+.In other words, \sqrt{\frac{2}{m} (C - V(x))} has a vertical tangent line at x = a. By the same reasoning - \sqrt{\frac{2}{m} (C - V(x))} also has a vertical tangnet line at x = a.We obtain that the graphs of \pm \sqrt{\frac{2}{m} (C - V(x))} connects to a smooth curve at x = a. The same conclusion holds at x = b.
- If V'(a) = 0, then V''(a) \le 0 (since V(x) < C in (a, b)). We have\lim_{x \to a+} \frac{d}{dx} \sqrt{\frac{2}{m} (C - V(x))} = \lim_{x \to a+} \frac{-V'(x)}{m\sqrt{\frac{2}{m} (C - V(x))}}.This limit is a \frac{0}{0} type that will be tedious to compute using L'Hò‚pital's rule. Instead, let us write\begin{aligned} V'(x) & = V'(a) + V''(a)(x - a) + O((x-a)^2) = V''(a)(x - a) + O((x-a)^2) , \\ V(x) & = V(a) + V'(a) (x - a) + \frac12 V''(a)(x - a)^2 + O((x - a)^3) \\ & = C + \frac12 V''(a) (x - a)^2 O((x - a)^3). \end{aligned}Using this approximation,\lim_{x \to a+} \frac{-V'(x)}{m\sqrt{\frac{2}{m} (C - V(x))}} = \lim_{x \to a+} \frac{ - V''(a)(x - a) + O((x - a)^2)} {m \sqrt{\frac{2}{m} (x - a)^2 (- \frac12 V''(a) + O(x - a))} } = \sqrt{m(-V''(a))}.In particular, if V''(a) < 0, we the tangent line of \sqrt{\frac{2}{m}(C - V(x))} has a positive slope.In this case, the graphs of \pm \sqrt{\frac{2}{m} (C - V(x))} should cross each other at an angle.
There is one particular case that we haven't discussed. If V(a) = C, V'(a) = 0, but V(x) > C for both x > a and x < a, then the level "curve" near the point x = a is a single point. (The discussion above does not apply since there is no interval on which C - V(x) > 0 near x = a).
The level curves can be sketched using the graphical method below, for the example
V(x) = \frac12 x^2 - \frac14 x^4.
Figure 2.
2.2. Equlibria and phase portrait of classical systems
The above discussed method for the phase portrait does not use any ODE theory, since we are only plotting the level curves of a function! Let us supplement it with linear analysis of the equilibria.
The equilibria of the system are of the type (x_0, 0), where x_0 satisfies V'(x_0) = 0. The only level sets \frac12 m y^2 + V(x) = C that contains an equilibria (x_0, 0) if
V(x_0) = C, \quad V'(x_0) = 0.
In other words, the graphs of y = V(x) and y = C must be tangent at x_0.
Proposition 2.4.
Suppose
V(x_0) = C, \quad V'(x_0) = 0.
Then- If V''(x_0) > 0, then the equilibrium (x_0, 0) is an elliptic centre.
- If V''(x_0) < 0, then the equilibrium (x_0, 0) is a saddle.
Example 2.5.
\ddot{x} + x - x^3 = 0.
Since F(x) = - x + x^3, we have
V(x) = \frac12 x^2 - \frac14 x^4.
The level curves of the energy \frac12 y^2 + V(x) can be sketched as before. There are three fixed points (-1, 0), (0, 0) and (1, 0). Since V''(0) > 0 and V''(\pm 1) < 0, (\pm 1, 0) are saddles and (0, 0) is a centre.
Example 2.6.
The pendulum
\ddot{x} + \frac{l}{g} \sin(x) = 0.
Example 2.7.
The double well
V(x) = - \frac12 x^2 + \frac14 x^4.
Example 2.8.
\dot{x} = y, \quad \dot{y} = - x - x^2.