---
title: "SU-MATH53 FEB052024"
source: https://www.jemoka.com/posts/kbhsu_math53_feb052024/
date: 2024-02-05
---

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:
\begin{equation} x&rsquo; = g(t,x, a) \end{equation}
if \(g\) is differentiable across \(t,x\) and \(a\), the IVP given by \(x&rsquo; = g(t,x,a)\) 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&rsquo; \(||x(t)||\) 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 \(a, y_0\), &ldquo;if I look at my solution sometime later, it would be a non-discontinuous change on the choice of initial condition&rdquo; Example Let&rsquo;s consider:
\begin{equation} y&rsquo; = -y \end{equation}
and take the initial value at:
\begin{equation} y(0) = y_0 \end{equation}
we have a solution such that:
\begin{equation} y(t) = y_0e^{-t} \end{equation}
which, at \(y(10)\), we obtain:
\begin{equation} y(10) = y_0e^{-10} \end{equation}
Which brings the question: &ldquo;how close should \(y_0&rsquo;\) be such that \(|y&rsquo;(10) - y(10)| \leq 10^{-5}\)?&rdquo;
We can recast this as:
\begin{equation} |y_0&rsquo; e^{-10} - y_0 e^{-10} | &lt; 10^{-5} \end{equation}
meaning:
\begin{equation} |y_0&rsquo; - y_0| &lt; \frac{10^{-5}}{e^{-10}} \approx \frac{1}{4} \end{equation}
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:
\begin{equation} \begin{cases} a \approx 10 \\ b \approx 28 \\ c \approx \frac{8}{3} \end{cases} \end{equation}
These attractors spins across two separate spheres, and the number of times the system spins around a particular area is unknown. It is called&hellip;
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:
\begin{equation} y&rsquo; = ry\qty(1-\frac{y}{k}) -h \end{equation}
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.
&ldquo;if the finish rate is too high over other parameters, you just die out.&rdquo;
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:
\begin{equation} x&rsquo;(t) = g(t,x(t)), x(t_0) = x_0 \end{equation}
For some interval \(t \in I\) where the IVP is satisfied; for any time interval \([t_1, t_2]\) inside \(I\) and any \(x_0&rsquo;\) near to \(x_0\), the associated \(x(t_0) = x_0&rsquo;\) should exist for the same interval \([t_1, t_2]\) and \(|| x&rsquo;(t) - x(t) ||\) is small for \(t\).
This extends for not just initial conditions, but also parameters as well. For function parameters \(a_0\) and \(a_0&rsquo;\).
Newtonian 3-body problem \begin{equation} m_1 x_1&rsquo;&rsquo; = \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&rsquo;t do the Cauchy Stability thing to it.
