One dimensional Flows

A flow on a set $X$ is a continuous function

$\phi:X \times \mathbb{R} \rightarrow X$

such that $\phi(x,0) = x$ and $\phi(\phi(x,t),s) = \phi(x,t+s)$ .

Usually in dynamical systems flows arise as solutions of differential equations, $y' = f(t,y)$ . However, some differential equations have solutions that diverge in finite time, or which have multiple solutions for the same initial conditions. Its important to have some understanding of when differential equations have well behaved solutions, so we will briefly review a couple of the main theorems about that.

Existence and uniqueness of solutions of ODEs

If we have a one-dimensional first order differential equation

$y' = f(t,y)$

with initial condition $y(t_0) = y_0$ for some given $t_0$ and $y_0$ , the most basic questions are whether there is a solution $y(t)$ , and if so whether this solution is unique.

The Peano existence theorem, proved by Giuseppe Peano around 1890, says that if the slope field $f(t,y)$ is continuous in an open set (e.g. an open disk in the $(t,y)$ plane) containing the initial condition $(t_0, y_0)$ , then there exists a solution $y(t)$ to the IVP in some interval of $t$ values containing $t_0$ .

The Picard-Lindelöf theorem

The Picard-Lindelöf theorem strengthens the Peano theorem: if the slope function $f(t,y)$ is both continuous and has a continuous partial derivative $\frac{\partial f}{\partial y}$ in an open set containing the initial condition, then there is a unique solution $y(t)$ to the IVP in some interval of $t$ values containing $t_0$ .

The proof of the Picard-Lindelöf existence and uniqueness theorem uses a solution approximation technique called Picard iteration. We start with the constant approximate $y_0(t) = y(t_0)$ . Then for each function $y_i(t)$ , we get a better approximate solution $y_{i+1}(t)$ from:

$y_{i+1}(t) = y_i(t_0) + \int_{t_0}^{t} f(s, y_i(s)) ds$

Example: local existence

Lets examine the ODE

$x' = x^2$

The implicit solution to this equation is

$\int x^{-2} dx = -1/x = \int dt = t + C_0$

which we can solve to find

$x = \frac{-1}{t + C_0}$

If we have a nonzero initial condition $x(0) = x_0$ , we can solve for $C_0 = -1/x_0$ , and then

$x = \frac{x_0}{1 - x_0 t}$

We can see that the solution becomes unbounded as $t$ approaches $1/x_0$ . Note that since $x^2$ is infinitely differentiable, the Picard-Lindelöf guarantees the existence of a unique solution near the initial data, but that solution may not be defined for all values of $t$ .

Example: failure of uniqueness

The following equation can be used to model the amount of water left in a draining bucket of water, if the bucket has a constant cross-sectional area:

$x' = -\sqrt{x}$

The right-hand side is infinitely differentiable for positive $x$ , but fails to be continously differentiable at $x=0$ .

An initial condition of the form $x(t_0) = 0$ has infinitely many solutions satisfying the ODE, corresponding to the fact that if the bucket is empty we can't tell when it was first empty. For any $t_e < t_0$ we can choose the solution $x=0$ for $t \ge t_e$ , and $x = (t-t_e)^2/2$ for $t < t_e$ .

Example: logistic ODE

This equation is one of the simplest nonlinear ODEs, and is used in a number of simple models, such as population models. It is usually called the logistic ODE:

$P' = P(1-P)$

The right hand side is a polynomial, and is infinitely differentiable for all $P$ , so there is a unique local solution for any initial value $P_0$ .

This can be solved directly by separation, solving

$\int \frac{1}{P(1-P} dP = \int dt$

with integration by parts, and then a moderate amount of algebra to solve for $P$ .

A clever alternative for this particular ODE is to invert the dependent variable with $N = 1/P$ ; then $N$ satisfies the linear ODE:

$N' = 1 - N$

which can be solved to get

$N = 1 + (N_0 - 1)e^{-t}$

and then

$P = \frac{P_0}{P_0 + (1 - P_0)e^{-t}}$

The logistic harvesting problem

Suppose a population $P$ is modeled by the logistic equation, but in addition this population (of fish, for example) is harvested at a certain rate $h$ . We can examine the location and stability of the equilibria as a function of $h$ without needing to explicitly solve the ODEs.

First we find the equilibria:

$P' = P (1 - P) - h = 0$

and from the quadratic formula we find that the equilibria $P_{\ast}$ are

$$P_{\ast} = \frac{1}{2} \left ( 1 \pm \sqrt{1 - 4h} \right ) $$

For $h < 1/4$ , there are two equilibria. At these two equilibria we can calculate the derivative of the slope field with respect to population:

$\frac{\partial }{\partial P} P'|_{P_*} = 1 - 2 P_* = \mp \sqrt{1 - 4h}$

and the sign of these derivatives means that the larger equilibrium is stable and the lower unstable. When the number or stability of the equilibria changes it is called a bifurcation. In this case we have a bifurcation at $h=1/4$ . If we plot this family of systems in the $h,P$ plane, we have a bifurcation diagram. An animated version for this system is shown below.

From flows to maps

There are several ways that ODEs and flows can be related to maps. The two most important are fixed-time maps, and Poincare maps.

To get a fixed-time map from a flow $\phi(y,t)$ we simply select a time $t_0$ and define the map to be $f(y) = \phi(y,t_0)$ .

A Poincare map is a little different, in that we select two subsets $S_0$ and $S_1$ of the state variable (denoted $y$ in most of our examples so far) that are transverse to the flow, and define a map between them. If $y \in S_0$ , then $f(y) = \phi(y,t_y)$ where $t_y$ is the smallest positive time such that $\phi(y,t_y) \in S_1$ .

Firefly synchronization

This example is oversimplified, but can be extended in very interesting ways to a higher-dimensional, but still tractable model of synchronization. This material is based on section 4.5 from Steven Strogatz's book, Nonlinear Dynamics and Chaos.

Suppose there is a single firely who, on its own will produce a flash at some frequency $\omega$ . We can model this by a first order differential equation,

$\dot{\theta} = \omega$

where $\theta$ is the phase angle for the firefly's periodic state.

Now suppose there is an external periodic signal (such as the coordinated flashing of a group of other fireflies) at frequency $\Omega$ , so that $\dot{\Theta} = \Omega$ , where the group's phase is $\Theta$ . If the single firefly wants to synchronize with the group, it can advance its phase state if it is a little behind, or slow its phase if it is ahead. One way to model this is:

$\dot{\theta} = \omega + A \sin(\Theta - \theta)$

where $A$ is a constant parameter representing how fast the firefly can adjust.

If we switch variables to focus on the difference between the phases, $\phi = \Theta - \theta$ , we can choose units of time and coupling strength to get the ODE

$\phi' = \mu - \sin \phi$

where $\mu = (\Omega - \omega)/A$ is the rescaled difference in frequency between the firefly and the global oscillation.

Exercises

Draw the phase portrait (the equilibria and the directions of trajectories) for the ODE

$y' = y^4 - y^2$