# Questions tagged [non-linear-dynamics]

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110 questions
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### Zero eigenvalue with deficient jacobian

Let $\dot{x} = f(x)$, $x \in \mathbb{R}^N$ be a system of ODEs with an equilibrium at the origin. Assume that the linearization at the origin has $K<N$ zero eigenvalues but only $K-1$ linearly ...
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### How do Static Loop-Transformations Work

Loop-Transformations (input feedforward or output feedback) can be used for passivity analysis, also in the case when considering a static (memoryless) nonlinearity. This is described for example in ...
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### In billiard systems, why are birkhoff coordinates needed to create area preserving maps?

Birkhoff co-ordinates, when used to obtain Poincaré sections of a billiards dynamics are often referred to as 'area preserving'.. why ?
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### An first integral of nonlinear differential equation as like forced pendulum nonlinear diff. eq.

I'm trying to face this nonlinear differential equation: $$y''(x)+\omega^2\sin\,y(x)=a\,x \,\;(1)$$ and I'm interested to found the solution of $y'(x)$ (an first integral) The homogeneous part of ...
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### Checking stability of a fixed point

If the fixed point is hyperbolic, then it is said that linearisation gives the correct result . Is there an intuitive way of understanding why this is so ? And for marginal cases, when the fixed ...
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### Phase trajectory must always enclose a fixed point

I found this problem in strogatz nonlinear dynamics . The theorem says, A closed phase space trahectory must enclose a fixed point . The question is asked as, is this true for phase surfaces other ...
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### Center manifold and projection onto center eigenspace

Consider a system $\dot{x} = f(x) \in \mathbb{R}^N$ with an equilibrium at $x_0$ for which the Jacobian has a zero eigenvalue and all other eigenvalues have negative real part. By the Reduction ...
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### Is chaos a topological property for continuous dynamical systems?

Following the definition of chaos given by Devaney, a continuous map $f$ on $(X,d)$ separable metric space with no isolated point is said chaotic if it is topologically transitive, that is for any ...
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### What are some good recommendations of nonlinear equations/functions?

I have a project for my matlab course. I need to find a nonlinear equation to use to find the roots of it using various root-finders. I then have to write up a paper talking about the these different ...
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### Revisit “example of an unstable fixed point for which the linearized dynamics are stable”

I am reading the following discussion: example of an unstable fixed point for which the linearized dynamics are stable The above discussion is for the vector field (continuous time). Is there an ...
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### Moser and Smale-Birkhoff homoclinic theorems

I have found in J. Moser "Stable and Random Motion in Dynamical Systems" the theorem about the topological conjugacy to the Bernoulli shift on a symbol space, and then again very well summarized and ...
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### Derivation of Polyanin's formula for Abel's ODE

There is a general solution from the Polyanin textbook for the equation, $y\cdot\frac{dy}{dx}-y=Ax+B$ The solution in parametric form is $x = C \cdot e^{-\int \frac {t \cdot dt}{t^2-t-A}}$ and ...
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### Describe discrete-time Langevin dynamics by its stationary distribution

Let's consider stochastic dynamics with discrete time step $t$ and states $x_t\in\mathbb{R}^d$ that evolve as $$\mathrm{p}(x_{t+1}|x_t)=\mathcal{N}(x_t+f(x_t), \Sigma).$$ Moreover, we assume the ...
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### System of nonlinear ODEs remains in positive cone

Given a system of three first order nonlinear ODEs \begin{align*} x_1' &= f(x_1,x_2,x_3) \\ x_2' &= g(x_1,x_2,x_3) \\ x_3' &= h(x_1,x_2,x_3), \end{align*} I'm hoping to ...
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### How to deal with non-equilibrium operating point

Given the nonlinear system \begin{align} \dot{x}_1 &= -4x_1 + 10x_2 + u \\ \dot{x}_2 &= -x_1 - 2x_2 - \log(1 + x_1^2) \\ y &= x_1 + x_2 \end{align} Assume the system should be ...
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### Phase portrait of a nonlinear system

I have an Lotka Volterra type of system as seen below; \begin{align} x' &= 10 x y -1/2 x - 1/10 x^2 \\ y' &= y - y^2 -x y \end{align} I would like to plot the phase portrait of the ...
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### linearizing dynamics about non fixed point for LQR implementation.

I am trying to implement LQR control for the cart pole system. I am curious if I can maintain a constant non-zero pole angle. So, I need to linearize my dynamics about my goal state. I know we can use ...
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### Differential equations - why we do care so much about (non)linearity?

This is a very simple question, but I feel I'm missing the bigger picture. Authors will talk of the horrors of nonlinear differential equations and that they're very difficult to solve, but why are ...
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### Does this dynamical system have another conserved quantity?

For the 3D system of ODEs: \begin{eqnarray}\dot{x} &=& -\beta x y \\ \dot{y} &=& \beta x y + \hat{\beta} z y - \delta y \\ \dot{z} &=& -\hat{\beta} z y + \delta y, \end{...