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Dane Sabo 2024-09-09 13:28:34 -04:00
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300s School/301. ME 2016 - Nonlinear Dynamical Systems 1/2024-09-09 Example.py Normal file
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300s School/301. ME 2016 - Nonlinear Dynamical Systems 1/2024-09-09 Frameworks and Review.md
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@ -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