# Questions tagged [non-linear-dynamics]

This tag is for questions relating to nonlinear-dynamics, the branch of mathematical physics that studies systems governed by equations more complex than the linear, $~aX+b~$ form.

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### Dynamics of the gain/attenuation $k(t)$ in the ODE-system

Given system of equation: \begin{cases} \frac{dx_1(t)}{dt}= \frac{df}{dx_1} \\ \frac{dx_2(t)}{dt}+x_2(t)=k(t) \cdot \frac{d^2f}{d^2x_1} \\ \frac{dk(t)}{dt}=??? \end{cases} where $x_1(t),x_2(t),k(t)$ - ...
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### Convergence proof of iterative linear quadratic regulator (iLQR)

Background The trajectory optimization problem can be expressed as: \begin{align} \min_{\mathbf{u}_{1}, \mathbf{u}_{1}, \ldots, \mathbf{u}_{T}} & \sum_{t=1}^{T} g(\mathbf{x}_{t}, \mathbf{u}_{t})\\ ...
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### How to know when a transcritical bifurcation occurs (Example 3.2.1 Strogaz Nonlinear Dynamics and chaos)

Picture of Question + solution Hi, In this question After we expand $\dot{x}$ around $x^*$ = 0 , we get $\dot{x}$ = (1-ab)x + ($\frac{1}{2}$a$b^2$)$x^2$ + O($x^3$). How do we jump from this to knowing ...
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### Nonlinear system with time-optimal control

Given nonlinear system: \begin{cases} \dot{x_1}=x_3+u \\ \dot{x_2}=-x_2+\dot{f} \\ \dot{x_3}=-x_3+x_2 \cdot \alpha \sin(\omega t) \\ \dot{x_4}=-x_4+x_2 \cdot (\frac{16}{\alpha^2}(\sin(\omega t)-\frac{...
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### Which way of solving from nonlinear control to choose?

I have a nonlinear system: \begin{cases} x'=f(x)+u \\ y=f(x) \end{cases} where $f(x)$ - gradient of some one-extremal function (for example $f=e^{-(x)^2}$), i.e. $\frac{df}{dx}$. Task: I want ...
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### Solve nonlinear, forced and damped Duffing oscillator

I solve Duffing equation with Van Der Paul's method: \begin{align} \ddot{x} + \omega^2 x + 2 \gamma \dot{x} + \beta x^3 = f \cos(\Omega t) \end{align} with $$x(t) = Re[A(t) \exp(i \omega_0 t)]$$ and ...
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### Does Poincaré-Bendixson theorem and Bendixon's criterion hold for this class of differential inclusions?

Poincaré–Bendixson theorem and Bendixson criterion are known to hold for dynamical systems $\dot{x} = f(x)$ in the plane (i.e. $x\in\mathbb{R}^2$). My question is: Are this results still valid in some ...
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### Uniqueness of a differential system with a linear subspace

I've encountered a system of differential equations, which takes the following form; \begin{equation} \begin{split} \dot{L}_1 &=\alpha L_1,\\ \dot{L}_2 &=\beta L_2,\\ \dot{N}& =\gamma N+\...