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300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-09-30 Limit Cycles.md

@@ -62,3 +62,35 @@ Therefore limit cycles are not possible.
 This is a method covered in the book. Sometimes is used to rule out limit cycles.
 ### Poincare - Bendixon Theorem.
 Book!
+# Perturbation Methods
+- Weakly nonlinear systems
+Linear Resonator:
+$m \ddot x + b \dot x + kx = f$
+Weakly Nonlinear:
+$m \ddot x + b \dot x + kx + \alpha x^3 = f$
+With a bookkeeping term:
+$m \ddot x + b \dot x + kx + \epsilon \alpha x^3 = f$
+
+## Asymptotic Expansion
+$x \neq x(t) \rightarrow x = x(t,\epsilon)$
+$x(t,\epsilon) = x_0(t) + \epsilon x_1(t) + \epsilon^2 x_2(t) + ... + \text{H.O.T.s}$
+Looking for solutions that are like
+$$x(t,\epsilon) ~ \sum_{k=0}^{\inf} x_k(t) \delta_c(\epsilon)$$
+Where $\delta$ is an asymptotically scaling number. This series sometimes doesn't converge but still gives useful information about the solution.
+**Example:**
+for $x>=0$
+$$\dot x + x - \epsilon x^2 = 0, x(0) = 2$$
+Develop a 3 term approximation using asymptotic expansion:
+$$x(t,\epsilon) = x_o(t) + \epsilon x_1(t) + \epsilon^2 x_2(t) + ...$$
+$$\dot x(t,\epsilon) = \dot x_o(t) + \epsilon \dot x_1(t) + \epsilon^2 \dot x_2(t) + ...$$
+Sub into the EOM:, and satisfy initial conditions $x_0(2) = 0; x_1(0) = x_2(0) = 0$
+$$  \dot x_o(t) + \epsilon \dot x_1(t) + \epsilon^2 \dot x_2(t) + x_o(t) + \epsilon x_1(t) + \epsilon^2 x_2(t) - \epsilon (x_o(t) + \epsilon x_1(t) + \epsilon^2 x_2(t))^2 = 0 $$
+Now that last term is going to yield higher order $\epsilon$ terms ($^2, ^4$). We can't get rid of these, we'll need to keep them. 
+Now collect terms:
+
+| Power | Expression |
+| ----- | ---------- |
+| $\epsilon^0$  | $\dot x_0 + x_0 = 0 \rightarrow x_0 = c_1e^{-t} \rightarrow x_0 = 2 e^{-t}$|
+| $\epsilon^2$ | $\dot x_1 + x_1 - x_0^2 = 0 \rightarrow \dot x_1 + x_1 - 4 e^{2t} = 0 \rightarrow x_1 = 4(e^{-t} - 2e^{-2t})$ |
+| $\epsilon^3$ | $\dot x_2 + x_2 -2(2e^{-t})(4 e^{-t} - e^{-2t}) \rightarrow ...$ | 
+Then we have an approximate solution for small $\epsilon$. What small means depends on the problem...