Ordinary Differential Equations: Limit Sets and Long Term Behavior
I wrote my Ordinary Differential Equations term project in LaTeX and thought it would be fun to see if I could convert it to markdown and post it here. Apparently MathJax makes this easy, so here it is!
Abstract
This paper will address the longterm behavior of systems of ordinary differential equations (hereafter ODEs), including the first and second dimensional cases. The specific behavior of interest is:
where ${y}(t)$ is a solution to a system of ODEs. Families of solutions will be considered analytically and numerically for behavior such as periodic orbits and fixed points. The considered problems will be plotted and have their limit sets enumerated.
Dimension D = 1
Consider the first order linear ODE:
By simply looking at the equation, it can be observed that it is autonomous (no independent variable is present in the ODE). From this we know that the behavior of the solution does not change with the independent variable $t$. Solving the differential equation through integration provides a solution:
Analytically it can be observed that with an initial condition of $y(0) = 0$ that $C_1 = 0$. Consequentially for any value of $t$, $y(t) = 0$:
For any other initial condition, it can observed that $y(t)$ will grow without bounds:
Because $y\prime = y$ is a one dimensional system, its phase plane only has a single dimension $y(t)$. The limit set includes all points in this dimension such that:
It directly follows that the limit set for the ODE $y\prime = y$ is $\{(0)\}$.
Dimension D = 2
Damped Vibrating Spring
Consider the second order system of linear ODEs:
This is derived from a second order unforced and damped harmonic motion ODE $y\prime\prime + 2cy\prime + \omega_0^2 = 0$ where $c = 1$ and $\omega_0 = 2$. When $c < \omega_0$ the system is overdamped. Knowing this, it should be expected that all solutions in the phase plane converge to a single point as the damping term overtakes the rest of the system. Consequentially, the limit set of the system contains the point of convergence for all solutions as $t\to\infty$.
Plotting the solution curves displays behavior typical of a spiral sink, and the trace determinant plane confirms it: $D(A) = 4,T(A) = 2$. Due to this behavior, all sollution curves in the phase plane converge to a single two dimensional equilibrium point $(0, 0)$ as $t\to\infty$. It follows that the limit set for this ODE is $\{(0, 0)\}$. While this limit set resembles the limit set of the one dimensional case, it should be noted that the ODE in the one dimensional case only had a non infinite solution as $t\to\infty$ with a single IVP, where as every IVP in the two dimensional case converged towards the origin.
Undamped Vibrating Spring
Consider the second order system of linear ODEs:
This is derived from a second order unforced and undamped harmonic motion ODE $y\prime\prime + 2cy\prime + \omega_0^2 = 0$ where $c = 0$ and $\omega_0 = 2$. When $c = 0$ the system is undamped. Knowing this, it should be expected that all solutions of $y(t)$ and $y\prime(t)$ to be repeating in nature, so they should form an ellipse like shape in the phase plane. Consequentially, the limit set of the system should contain as many members as there are solutions to initial value problems. A plot confirms these suspicions:
This system clearly exhibits a center behavior, which is confirmed by the trace determinant plane: $D(A) = 4,T(A) = 0$. It follows that the limit set contains every curve in the phase plane formed by the general solution of the ODE and its derivative:
It should also be noted that the origin of the phase plane $(0, 0)$ is in the limit set. This can easily be verified by solving the initial value problem for $y(0) = 0, y\prime(0) = 0$. Finally, the limit set for this ODE can be characterized by the two behaviors mentioned above:

$(y=0,y\prime=0)$: Solutions that start at the origin stay at the origin as $t\to\infty$

$y\ne0,y\prime \ne 0$: Solutions that start outside of the origin stay in a periodic solution curve as $t\to\infty$ which is defined by the system solution for the initial value problem.
Undamped Pendulum
Consider the second order system of nonlinear ODEs:
This is derived from the undamped pendulum ODE $\theta\prime\prime = sin(\theta)$. There are quite a few initial conditions $p_0$ to consider.

For $p_0\in\{ (\theta = 2n\pi, \omega_0 = 0) n\in\mathbb{Z} \}$, the phase plane solution should be a single point $p_0$ as the pendulum has no energy.

For $p_0\in\{\theta_0 \ne 0, \omega_0 = 0\}$ the phase plane solution should look like a center around a point $p_c \in \{ (\theta = 2n\pi, \omega) n\in\mathbb{Z} \}$ as the pendulum converts potential energy to velocity, back to potential energy, and finally changing directions to repeat the process.

For $p_0\in\{(\theta_0=0,\mid\omega_0\mid \lessapprox 2.00027)\}$ , the pendulum should behave like case two.

For $p_0\in\{(\theta_0=0,\mid\omega_0\mid \gtrapprox 2.00027)\}$, the pendulum no longer changes direction and $\lim_{t\to\infty} \theta(t) = \pm\infty$

For $p_0\in\{ (\theta_0 = \pi + 2n\pi, \omega_0 = 0) n\in\mathbb{Z} \}$, the phase plane solution should be a single point $p_0$ as the pendulum needs a nudge in either direction to start moving.
The plots above confirm the predicted behavior of solutions. The limit set can be broken down into three different cases:

Solutions where $\omega$ contains positive and negative values behave as a center around some point $p_c \in \{(\theta_0 = 2n\pi, \omega_0 = 0)  n\in\mathbb{Z} \}$. This is confirmed by looking at the trace determinant plane for the center equilibrium: $D(J)=1, T(J)=0$.

$p_0 \in \{(\theta_0 = 2n\pi, \omega_0 = 0)  n\in\mathbb{Z} \}$: Solutions to this IVP stay at the angle they started at as $t\to\infty$. These equilibrium points are shared with case one.

$\{ (\theta_0 = \pi + 2n\pi, \omega_0 = 0)  n\in\mathbb{Z} \}$: Solutions to this IVP stay at the angle they started at as $t\to\infty$. These equilibrium points are saddle points in the trace determinant plane: $D(J)=1, T(J)=0$.
It is important to keep in mind that solutions where $\lim_{t\to\infty} \omega(t) \neq\pm\infty$ are not a member of the limit set if $\lim_{t\to\infty} \theta(t) = \pm\infty$. This excludes any solutions for which $\omega$ is exclusively $> 0$ or $< 0$ as $t\to\infty$. However, we can still make a definitive statement as to the behavior of $\omega$ as $t\to\infty$.
Damped Pendulum
Consider the second order system of nonlinear ODEs:
This is derived from the damped pendulum ODE $\theta\prime\prime = sin(\theta)  c\theta\prime$. A reasonable guess as to its behavior would be behaving much like the undamped case except in the cases that resulted in endless oscillation. On the phase plane, these cases should converge to $(\theta_0 = 2n\pi  n\in\mathbb{Z}, \omega_0 = 0)$ instead.
Based on analysis of the system and observed behavior of the numerical plot, a limit set can be constructed:

When $\{(\theta_0 = \pi + 2n\pi, \omega_0=0) n\in\mathbb{Z}\}$, solutions remain where they started. This can be verified because these values are in the set of equilibrium points. The trace determinant shows saddle point behavior for the set of initial conditions: $D(J)=1, T(J)=\frac{1}{2}$

All solutions not included in case one converge to a spiral sinks in the set of $\{(\theta = 2n\pi, \omega = 0) n\in\mathbb{Z}\}$ depending on initial conditions. This can be confirmed by recognizing that equilibrium points which follow the same pattern are all in the spiral sink region of the trace determinant plane: $D(J)=1, T(J)=\frac{1}{2}$
Competing Species
Consider the second order system of nonlinear ODEs:
In a competing species system, the populations of two species interact with each other and/or themselves. To analyze such a system for long term behavior, the equilibrium points should first be considered:
Solving for the xnullcline, ynullcline, and nullcline provides the set $\{(0, 0), (1, 0), (0, \frac{4}{7}), ({\frac{3}{5}, \frac{2}{5}})\}$ Next, the system is linearized by computing the Jacobian:
Next the Jacobian is used to examine the equilibrium in the trace determinant plane:
Plotting for several initial value problems, a few things can be observed:

When $x$ and $y$ have an initial values $> 0$, their solutions gravitate towards $\frac{3}{5}$ and $\frac{2}{5}$. This is consistent with the predicted nodal sink behavior.

Initial values of $x$ and $y$ that start at zero stay at zero, which is expected when starting at the center of a nodal source.

When $y = 0$, Initial values of $x > 0$ gravitate towards one. This makes sense when $y = 0$ reduces the first equation to $x\prime = x(x  1)$ which clearly has a root of $1$. It follows that it has an equilibrium point there as well. Examining the trace determinant of $(1, 0)$ places it in the region of saddle points.

When $x = 0$, $y$ gravitates towards $\frac{4}{7}$. This makes sense considering the second equation is reduced to $y\prime = y(4  7y)$, which has a root $\frac{4}{7}$. Examining the trace determinant of this point it clearly falls into the region of saddle points.
Considering the above, the limit set contains individual points on the phase plane: $\{(0, 0), (1, 0), (0, \frac{4}{7}), ({\frac{3}{5}, \frac{2}{5}})\}$. This system contains more non periodic individual points in its limit set than any previously examined system while containing no center curves.
Van der Pol’s Equation
Consider the following system of nonlinear ODEs:
To analyze this system, first the equilibrium points need to be solved using the intersection of the nullclines:
Next, the system is linearized and the trace determinant of the equilibrium point computed:
The equilibrium point is a special case with real eigenvalues and $T^2 = 4D$. Plotting the phase plane reveals atypical behavior:
There are two clear behaviors here, both of which are clearly members of the limit set:

For $(x = 0, y = 0)$, the solution does not leave the origin of the phase plane.

All other initial values seem to converge into the exact same periodic closed curve. This is different than previous center periodic solutions where there were infinitely many different paths depending on initial conditions.
Setting the initial condition to a part of the closed loop yields the following result:
Extra Work
Approximating Van der Pol’s Solution Curve
While I wasn’t able to find a solution to Van der Pol’s equation, I was not content walking away without at least approximating a solution. Here are the steps I took to approximate a curve. First, we apply a rotation using a linear transformation with $\theta=\frac{\pi}{16}$ (Matrix $A$ is the output of ODE45)
Next a shear transform is applied with $k = 0.03$:
A polynomial fit is found with $n = 26$. Next we apply the inverse of the linear transformations previously applied.
The moment of truth:
While the fit may not be anywhere near perfect, it is a good first attempt and perhaps worthy of more exploration at a later time.