vault backup: 2024-09-09 13:28:34

.obsidian/workspace.json

@@ -143,7 +143,8 @@
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@@ -203,7 +206,6 @@
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300s School/301. ME 2016 - Nonlinear Dynamical Systems 1/2024-09-09 Frameworks and Review.md

@@ -41,6 +41,23 @@ The system is at equilibrium where $\frac{dx}{dt} = 0$. It won't move from this
 Is this system stable? Check the eigenvalues of A.
 ### Nonlinear Systems
 **Recall: $\dot{x} = 1-2\cos x$**
-#### The Phase Plane
+We can qualitatively describe systems using the phase plane:
+*Insert graphics from class*
 
+How is this useful to us engineers?
+We are going to see systems that are nonlinear, and they can give us ideas about where things could blow up. In our second example, we have generally a pretty safe area below x = 2. Anywhere below there, we know we're going to end up at x = -2, but above x =2, all hell breaks loose.
 
+This is what we care about. We want to know where in our nonlinear system domains things can become dangerous.
+## How do we numerically get a time domain response?
+Numericaly:
+$$ \dot x = f(x) $$
+$$\frac{dx}{dt} = \lim_{\Delta t \rightarrow 0} \frac{f(x(t+\Delta t))-f(x(x))}{\Delta t} $$
+This is the tangent (or the secant while $\Delta t =/ 0$)
+>[!note] Eulers Method
+>Therefore, for finite $\Delta t$:
+> $$ f(x(t+\Delta t)) = \frac{dx}{dt} \Delta t +  f(x(t))$$
+> Limitations: innaccurate if time steps are large.
+> There are better methods!
+> ode45() <- Variable Step Runge-Kutta 
+
+We're going to use a lot of odeint in SciPy