1. Basic theory of nonlinear systems
A general ODE on \R^n is given by the equations
\begin{cases} \dot{x}_1 = f_1(x_1, \cdots, x_n), \\ \cdots \\ \dot{x}_n = f_n(x_1, \cdots, x_n). \end{cases}
In vector form, it's often convenient to write
\dot{\bx} = \bfF(\bx),
here \bfF: \R^n \to \R^n is a vector field. A vector field can be visualized by a field of arrows on the space. A function \bx(t) is a solution if the velocity vector \dot{\bx}(t) is always equal to the vector field at the point \bx(t) (which is \bfF(\bx(t))).
The space \R^n of on which the ODE is defined is often called the phase space.
Remark 1.1.
When we are considering a higher order equation, say
\ddot{x} + \dot{x} + x = 0,
The corresponding first order system is given by y = \dot{x} and
\dot{x} = y, \quad \dot{y} = - x - y.
In this case the phase space always refer to the space \R^2 (i.e. after conversion to first order systems).
The following theorem is the foundation for our geometrical understanding of ODE:
Theorem 1.2. (Existence and uniqueness theorem).
Suppose the vector field
\bfF(x)= \bmat{f_1(x_1, \cdots, x_n)\\ \vdots \\ f_n(x_1, \cdots, x_n)}
is a C^1 function. (This mean each of f_1, \cdots, f_n are C^1 functions, which in turn means that all partial derivatives exists and are continuous functions).
Then given any \bx_0, there exists \epsilon > 0 and a unique solution \bx(t), t \in [-\epsilon, \epsilon] so that
\dot{\bx}(t) = \bfF(\bx(t)), \quad t \in (-\epsilon, \epsilon), \quad\bx(0) = \bx_0.
It's helpful to reinterpret this theorem in terms of what it means for the phase portrait.
(Existence of solution). On every point \bx_0 of the space \R^n, there exists a (short) smooth curve parametrized by \bx(t), t \in [-\epsilon, \epsilon] that is everywhere tangent to the vector field.
(Uniqueness of solution). The solutions curves to the ODE cannot intersect.
The fact that the existence and uniqueness theorem only guarantees a "short" solution curve may look strange at first, but in fact we can always extend the solution to a large interval, except in one special case.
(Extension of solution): Given a "short" solution \bx(t) defined on [-\epsilon, \epsilon], one can apply the existence of solution for \bx_0 = \bx(\epsilon) to extend the solution to [-\epsilon, 2\epsilon]. This extension can be applied indefinitely unless \bx(t) \to \infty.
These statements ensure that we can find solution curves at every point in the space \R^n. Then extend them to long solution curves, unless they travel so fast that they zoom to infinity. We will revisit the latter case after discussing some examples in one-dimensional systems.
2. Phase portrait of 1-dimensional nonlinear systems
In this section we consider the ODE
\dot{x} = f(x),\qquad (2.1)
where x \in \R. The phase portrait is given by the vector field on a line. Because the solution curves / trajectories cannot cross each other, this puts a great deal of limitations possible phase portrait.
When analyzing a system, the general rule is to first look for "timeless" features of the system. The simplest timeless feature is the equilibrium.
x_0 is an equilibrium of the system if f(x_0) = 0. If x_0 is an equilibrium, then the constant function x(t) \equiv x_0 is a solution of the ODE.
The equilibrium is sometimes also called a fixed point.
Because the trajectories cannot cross, equilibria serve as barriers in the phase space. (Only true in dimension 1 though!) Moreover, in every interval in between the equilibria, the vector field f(x) cannot change sign (otherwise there would be an equilibrium in the interval), therefore all trjectories moves from one end point to the other.
Example 2.1.
f(x) = x + \sin(x).
Example 2.2. (The logistics equation).
A simple model of population growth is \dot{x} = rx, which models unlimited growth under the reproduction rate r> 0. A more realistic model puts an upper restriction on the population, denoted K > 0, so that the closer the population is to the cap, the harder it is to grow. This leads us to the equation
\dot{x} = rx\left( 1 - \frac{x}{K}\right).
Example 2.3.
f(x) = \sin^2(x).
Definition 2.4.
An equilibrium x_0 in a 1-dimensional system is called stable if there is \epsilon > 0 such that f(x) < 0 for x \in (x_0, x_0 + \epsilon) and f(x) > 0 for x \in (x_0 - \epsilon, x_0).
The equilibrium x_0 is called unstable if there is \epsilon > 0 such that f(x) > 0 for x \in (x_0, x_0 + \epsilon) and f(x) < 0 for x \in (x_0 - \epsilon, x_0).
The equilibrium x_0 is called half-stable if there is \epsilon > 0 such that f(x) has the same signs for x \in (x_0, x_0 + \epsilon) and x \in (x_0 - \epsilon, x_0).
Exercise 2.5.
Classify all the equilibriain all the examples.
Not all equilibria, even for 1-dimensional system, can be classified into one of these three types.
Example 2.6.
f(x) = x^2 \sin \frac{1}{x}.
2.1. Linear stability analysis
If x_0 is an equilibrium, an easy way to get information about the equilibrium is by studying the derivative f'(x_0) of the vector field at x_0. Since the differential of a function is the linear approximation of the function, this is often called the linear analysis at x_0.
Proposition 2.7.
If f'(x_0) > 0, then x_0 is unstable. If f'(x_0) < 0, then x_0 is stable. However, nothing definitive can be said if f'(x_0) = 0.
Proof.
Suppose f'(x_0) > 0. Since
f'(x_0) = \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0},
for x close to x_0 we have
\frac{f(x) - f(x_0)}{x - x_0} > 0.
This means f(x) - f(x_0) > 0 if x > x_0 and f(x) - f(x_0) < 0 if x < x_0, implying x_0 is unstable.
The case f'(x) < 0 is similar, we leave it as an exercise.
If f'(x_0) = 0, we cannot draw any conclusion about the sign of f(x) - f(x_0) in general.
In 1-dimensional system, it's often possible to identify the sign of f(x), which provides more information than linear analysis. However, in higher dimension, there is no easy way to draw the phase portrait, hence linear analysis becomes more valuable.
2.2. Counter examples
The existence and uniqueness theorem requires the vector field to be C^1. The following example shows that uniqueness can fail if the vector field does not have continuous derivatives.
Example 2.8.
f(x) = x^{\frac13}.
Our second counter example concerns the continuation of solutions. If f(x) is a C^1 vector field, then the trajectory is a curve x(t) parametrized by t. However, it's not clear that x(t) is defined for all t \in \R. The "bad case" is that the vector field is accelerating the trajctory so much that the trajectory go to infinity in a finite amount of time.
Example 2.9.
f(x) = x^2.
Definition 2.10.
A system for which all trajectories are defined for all time is called complete.
Remark 2.11.
Note that even if a system is not complete, one can still draw its phase portrait. Completeness is about how fast the orbits move along the vector field.
After introducing the concept of completeness, we introduce the concept of the flow of a system.
Definition 2.12.
Suppose the system
\dot{\bx} = \bfF(\bx), \quad \bx \in \R^n
is complete. For any \bx_0 \in \R^n, we define \phi(t; \bx_0) to be the solution \bx(t) satisfying the initial condition \bx(0) = \bx_0.
We visualize \phi(t; \bx_0) as the position where a particle starting at \bx_0 will be, after "flowing" along the vector field for time t.
2.3. Bifurcation in 1-dimensional systems
A bifurcation analysis is to study how the structure of the system changes based on an external parameter.
Example 2.13.
Consider the system
\dot{x} = r + x^2,
where r \in \R is a parameter.
The equilibria of the system are given by the solutions of the equation
r + x^2 = 0.
There are 3 cases:
r < 0. The equilibria are x = \pm \sqrt{-r}, and f(x) takes the signs +, -, + in the intervals (-\infty, -\sqrt{-r}), (-\sqrt{-r}, \sqrt{-r}), and (\sqrt{-r}, \infty).
r = 0. One equilibrium at x = 0, but f(x) > 0 on both sides of the equilibrium.
r > 0. There are no equilibria, f(x) > 0 throughout.
One common way is to depict the fixed point and there stability with the parameter r as the horizontal axis. The stable fixed point is depicted with a solid curve, while the unstable fixed point is depicted with a dashed curve. This is a version of bifurcation diagarm. This particular type of bifurcation is called saddle-node bifurcation.
Figure 1.
Example 2.14.
Consider the system
\dot{x} = rx - x^3.
We can perform a similar analysis for different values of r. This bifurcation is one of the two subtypes of the pitchfork bifurcation.
Figure 2.
3. Study systems using poloar coordinates
Many 2-dimensional system simplify significantly after converting to polar coordinates. Suppose x(t), y(t) is a solution of the system
\begin{cases}\dot{x} = f(x, y),\\\dot{y} = g(x, y), \end{cases}\qquad (3.1)
we define the function r(t), \theta(t) from x(t), y(t) by solving the equation
x = r \cos \theta, \quad y = \sin \theta.\qquad (3.2)
Remark 3.1.
While the formula \tan \theta = \frac{y}{x} is commonly used, this is not an accurate way to define the coordinate change. First of all, this formula is undefined if either x = 0 or \theta = \frac{\pi}{2}. Secondly, \tan \theta = \frac{y}{x} does not uniquely determine an angle \theta \in [0, 2\pi). Therefore, I do not recommend using this formula.
To rigorously derive the polar equation, we plug (3.2) into the equation (3.1). Apply chain rule, and noting that both r and \theta are functions of t, we get
\begin{cases} \dot{r} \cos \theta - r \sin \theta \cdot \dot{\theta} = f(r \cos \theta, r \sin \theta), \\ \dot{r} \sin \theta + r \cos \theta \cdot \dot{\theta} = g(r \cos \theta, r \sin \theta).\end{cases}
Treating \dot{r}, \dot{\theta} as unknowns, we can solve the systems of 2 linear equations to get the expression for r and \theta.
Example 3.2.
Skecth the phase portrait of the equations
\begin{cases}\dot{x} = y + x(1 - x^2 - y^2), \\\dot{y} = - x + y(1 - x^2 - y^2).\end{cases}
After substitution, we get
\begin{cases} \dot{r} \cos \theta - r \sin \theta \cdot \dot{\theta} = r \sin \theta + r \cos \theta (1 - r^2) \\ \dot{r} \sin \theta + r \cos \theta \cdot \dot{\theta} = - r \cos \theta + r \sin \theta (1 - r^2).\end{cases}
Let's denote by (1) the first equation and (2) the second equation. Compute (1) \times \cos \theta + (2) \times \sin \theta and simplify (try it yourself!), we get
\dot{r} = r(1 - r^2).
To find \dot{\theta}, we compute (2) \times \cos \theta - (1) \times \sin \theta, and simplify (again, do the calculations!) to get
\dot{\theta} = - 1.
In the polar equation, the variables separate. We can plot the phase portrait for the r variable, then keep in mind that the \theta variable will decrease at a constant speed (in Cartesian coordinates, the vector rotate clockwise with constant speed).
We can "automate" the way to solve \dot{r} and \dot{\theta} in the following way. Since
\begin{aligned} \dot{x} & = \dot{r} \cos \theta - r \sin \theta \cdot \dot{\theta}, \\ \dot{y} & = \dot{r} \sin \theta + r \cos \theta \cdot \dot{\theta}.\end{aligned}
Compute (1) \times \cos \theta + (2) \times \sin \theta to get
\dot{r} = \dot{x} \cos \theta + \dot{y} \sin \theta.
Compute (2) \times \cos \theta - (1) \times \sin \theta to get
r \dot\theta = \dot{y} \cos \theta - \dot{x} \cos \theta.
We can use the equations for \dot{x} and \dot{y} (after substituting x = r \cos \theta, y = r \sin \theta) to get the equations for \dot{r} and \dot{\theta}.
Example 3.3.
\begin{cases} \dot{x} = x - y - x(x^2 + y^2) + \frac{xy}{\sqrt{x^2 + y^2}},\\ \dot{y} = x + y - y(x^2 + y^2) - \frac{x^2}{\sqrt{x^2 + y^2}}.\end{cases}
4. Limit sets
In this section, we assume that
\dot{\bx} = \bfF(\bx)
is complete (see Definition 2.10), hence the flow\phi(t; \bx_0) is defined (see Definition 2.12).
Definition 4.1.
A point \by_+ is an \omega-limit point of \bx_0 if there exists an infinite sequence t_j, such that t_j \to \infty and \phi(t_j; \bx_0) \to \by as j \to \infty.
A point \by_- is an \alpha-limit point of \bx_0 if there exists an infinite sequence t_j, such that t_j \to - \infty and \phi(t_j; \bx_0) \to \by as j \to \infty.
Example 4.2.
Identify all the possible \omega and \alpha-limit points in Example 3.2.
Definition 4.3.
A set D \subset \R^n is forward invariant if for all \bx_0 \in D, \phi(t; \bx_0) \in D for all t > 0.
A set D \subset \R^n is backward invariant if for all \bx_0 \in D, \phi(t; \bx_0) \in D for all t < 0.
A set D is invariant if it is both forward and backward invariant.
