In this course, we will be focus on qualitative properties of continuous
and discrete dynamical systems, complementing other common methods, like explicit
solutions and numerical approximations. Explicit solutions, even available
in certain cases, may not be useful. For example, the general solution of the
system \begin{equation} \label{mysys}
\frac{\mathrm{d} x}{\mathrm{d} t} = xy, \quad \frac{\mathrm{d} y}{\mathrm{d}
t} = \frac{1-x^2+y^2}{2} \end{equation}
with initial condition \(x(0)=x_0,y(0)=y_0\) is
\begin{align*} x(t) &= \frac{2x_0} {1+x_0^2+y_0^2+(1-x_0^2-y_0^2)\cos t - 2y_0\sin
t},\cr y(t) &= \frac{2y_0\cos t + (1-x_0^2-y_0^2)\sin t} {1+x_0^2+y_0^2+(1-x_0^2-y_0^2)\cos
t - 2y_0\sin t}. \end{align*}
Click your mouse on the plane below, and the solution will move. Instead of solving
the solution given above, we are more interested in properties like
Is there point that does not move (so-called stationary or fixed point?
Can we find the equations for the trajectories?
What are the long time behaviours, periodic, converging to some points,
or escaping to infinity?
Before considering complicated nonlinear systems, we start with a few basic notations
and concepts in the next section.
Exercise.
The system \eqref{mysys} can be solved by
introducing a new variable \(z\) which satisfies \(\dot{z}=−yz\)
with the initial condition \(z(0)=1\).
Show that \(xz\) is conserved, to deduce that \(xz=x_0\).
Substitute \(x=x_0/z\) and \(y = -\dot{z}/z\) into the equation \(\dot{y}=\frac{1-x^2+y^2}{2}\)
to get a second order ODE for \(z\).
From the second order ODE for \(z\), show that \(\frac{\dot{z}^2+z^2+x_0^2}{z}\)
is invaraint, and hence
\[
\frac{\dot{z}^2+z^2+x_0^2}{z} = 1+x_0^2+y_0^2.
\]
Here you need the equation \(\dot{z} = -yz\) to get \(\dot{z}(0)=-y_0\).
The solution of the above first order ODE
\( \frac{\dot{z}^2+z^2+x_0^2}{z} = 1+x_0^2+y_0^2\) is
\( z(t) = \frac{1+x_0^2+y_0^2}{2} + A\cos t + B\sin t\), for some constants \(A\) and \(B\).
Use the initial condition for \(z(0)\) and \(\dot{z}(0)\) to find the solution \(z(t)\), and then
\(x(t),y(t)\).
Exercise
Another way to solve the system \eqref{mysys} is to introduce the complex variable
\[
w(t) = x(t) + iy(t).
\]
Show that \(w(t)\) satisfies the first order ODE (which is separable!)
\[
\dot{w} = \frac{1-w^2}{2}i,\qquad
w(0) = x_0 + iy_0,
\]
and hence obtain the solution.