Sensitivity to Initial Conditions + Parameters. ODE Existence and Uniqueness We can recast all high order systems into a first-order vector-valued system. So, for any system:

if g is differentiable across t,x and a, the IVP given by x’ = g(t,x,a) and x(0) = x_0, has the property has that: the ODE has a solution x(t_0) = x_0 for any t_0, and any two solutions on the interval coincide as the same solution The only way for a solution to fail to extend temporally is due to the bounds’ ||x(t)|| becomes unbounded as t approaches the endpoints On any interval t_0 \leq t \leq T the solution y_{a,y_0} depends continuously on a, y_0, “if I look at my solution sometime later, it would be a non-discontinuous change on the choice of initial condition” Example Let’s consider:

and take the initial value at:

we have a solution such that:

which, at y(10), we obtain:

Which brings the question: “how close should y_0’ be such that |y’(10) - y(10)| \leq 10^{-5}?” We can recast this as:

meaning:

If you flip it over, you will have extreme instability. Example \begin{equation} \begin{cases} \dv{x}{t} = a(y-x) \ \dv{y}{t} = (b-z)x-y \ \dv{z}{t} = xy-cz \end{cases} \end{equation} this seems innocuous, but no. If we set our parameters to be weirdly specific values:

These attractors spins across two separate spheres, and the number of times the system spins around a particular area is unknown. It is called… Deterministic Chaos Deterministic Chaos is a hard problems which there is a bounded region in which the behavior happens, but the system is bounded. Another Example Logistic expression:

You can get solutions of this form for some carrying capacity k and a constant rate of removal h. You can observe that we can build a phase line of this system, and observe. This behavior is called bifurcation: when some h is high enough, our whole system dies out. “if the finish rate is too high over other parameters, you just die out.” You can also draw a plot, where the x axis is some parameter p, and phase plot can be drawn sideways. Cauchy Stability Suppose x(t) satisfies:

For some interval t \in I where the IVP is satisfied; for any time interval [t_1, t_2] inside I and any x_0’ near to x_0, the associated x(t_0) = x_0’ should exist for the same interval [t_1, t_2] and || x’(t) - x(t) || is small for t. This extends for not just initial conditions, but also parameters as well. For function parameters a_0 and a_0’. Newtonian 3-body problem \begin{equation} m_1 x_1’’ = \frac{-Gm_{1}m_2}{|x_1-x_2|^{2}}- \frac{Gm_{1}m_3}{|x_1-x_3|^{2}} \end{equation} you will note that this expression has no close form solution, so you can’t do the Cauchy Stability thing to it.