Questions tagged [nonlinear-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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Classifying a second order non-linear ODE

I am currently dealing with the following ODE as a stationary, special case version of a PDE model derived from Kuramoto-Sivashinsky. $$y'' y' = ay$$ Where $a$ is a real (constant) parameter. I am ...
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How to best write a sum of chains

Disclaimer: I know hardly anything about this math topic, I don't even know if what I will describe can be called a "set of chains", I am asking this question precisely to get some advice on ...
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L2-preserving discretization of inviscid Burgers’ equation

I’m looking for a stable discretization of the inviscid Burgers’ equation that exactly preserves the L2-norm of the solution. Does such a discretization exist? I’d appreciate any insight/references!
55 views

Requirements for invertibility of $A B A^T$ in constrainted dynamics

What are the requirements for matrix $A$ (that isn't a square matrix), so that the matrix $A B A^T$ is invertible, given that $B$ is non-singular? Some details for the matrices: $B$ is the $n \times n$...
1 vote
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2k views

What is meant when mathematicians or engineers say we cannot solve nonlinear systems?

I was watching a video on "system identification" in control theory, in which the creator says that we don't have solutions to nonlinear systems. And I have heard this many times in many ...
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1 vote
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Is it possible to find a solution to ODEs assuming the solution is periodic with known period?

I have a nonlinear system of ODEs with known constant coefficients $A, B, C, D, E, F, M$: \begin{align} &\dot{n}(t)=-An(t)+Bm(t)n(t)+Cm(t) \\ &\dot{m}(t)=-Bm(t)n(t) + (M-m(t))R_0 \sin{\omega t}...
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Find all the values of r in this situation ( Nonlinear Dynamics)

Find all the values of r so that the equation dx/dt=cos(rx) defines a vector field on the circle. My answer is that ; By the definition of a vector field on the circle, dx/dt=cos(rx) must be real ...
85 views

numerically solving for the fixed points of a system of nonlinear ODEs

I was looking at an excellent lecture series on Robotics by Russ Tedrake, and he discusses Linear Quadratic Control (LQR) for system of nonlinear differential equations. So as he suggests, robots are ...
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Is a system, that is globally asymptotically stable for any constant input also input-to-state stable? [closed]

I am referring to the ISS definition by Sontag of ${\displaystyle |x(t)|\leq \beta (|x_{0}|,t)+\gamma (\|u\|_{\infty }).}$ I understand that 0-GAS is a necessary condition for ISS. But is GAS for all ...
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Are there any examples of diffusion PDEs with nonlinear complications, that would possess analytical solutions?

I need an example (at least one, but more are welcome) of nonlinear PDEs in one space dimension (finite interval), containing transient diffusional terms plus some nonlinear complications, with ...
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Are there general solutions to quadratic, 2D, continuous, time-invariant dynamical systems?

I am a bit new to dynamical systems and don't know my way around terminology, so have had a hard time answering this for myself. I know the basics of theory for 2D linear, time-invariant systems, i.e.,...
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How to calculate the monodromy matrix for a system of nonautonomous nonlinear differential equations?

I am interested in analyzing the stability of the periodic orbits resulting from the Van der Pol system periodically perturbed by a time-dependent external forcing. Mathematically it would be the ...
67 views

Final value of a recursion

Problem Given $p_1, \sigma > 0$, consider the following recursion \begin{equation*} p_{i}=(1-L_i)p_{i-1} \qquad i=2,\dots,k \end{equation*} where \begin{equation*} L_i \triangleq \frac{p_{i-1}}{p_{...
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Singularity of a non- linear second order ODE

I have the encountered a singularity in the equation below . $$y^{\prime \prime}(x)+\frac{2}{x} y^{\prime}+\left[y-\left(1+\frac{2}{x^2}\right)\right] y(x)=0, \quad 0<x<+\infty,$$ with ...
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Radially bounded Lyapunov function and global stability

I came accross this link about the necessity of the Lyapunov function being radially unbounded. My understanding is that this condition is unnecessary if the time derivative along solution ...
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1 vote
106 views

Non vanishing gradient condition in control barrier funcions.

I am reading about barrier functions in control engineering/dynamical systems. These tools are used to prove that the system is forward invariant with respect to a set $\mathcal{C}$ (i.e., starting in ...
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Exponential Stability and Lasalle's Invariance Theorem

It is well known that a system $\dot{x}=f(x)$ with $x \in \mathbb{R}^n$ is exponentially stable if there exists a Lyapunov function $V(x)$ which satisfies \begin{align} k_1\Vert x \Vert \leq V(x) &...
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Convergence of a dynamics with rate multiplier for coordinates

We are given a continuous dynamics $x(t) \in \mathbb{R}^n_{> 0}$ that follows $\frac{dx}{dt} = g(x)$, where $g : \mathbb{R}^n_{> 0} \rightarrow \mathbb{R}^n$ is a smooth continuous function. We ...
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from local stability to global stability

Suppose I have the system $x'=F(x)$ with $F:\mathbb{R}^n\rightarrow\mathbb{R}^n$. I denote by $J(x)$ the Jacobian matrix, that is, $J_{ij}(x)=\partial F_i/\partial x_j (x)$. Suppose I know that for ...
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How to visualize low-dimensional torus in a high-dimensional system?

I have a system of very high-dimensions (1000s of independent variables), but I could show that the dynamics is attracted to a 1D limit cycle or a 2D torus (with commensurate frequencies, so still ...
79 views

Are two separate limit cycles in a dynamical system possible

In all the examples I've seen before with two limit cycles, the limit cycles are always concentric (there is an unstable point at center, a stable limit cycle on the middle and an unstable limit cycle ...
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1 vote
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Realizing a modified transport equation

Stated somewhat informally, the continuity equation or transport equation $\partial_t\rho_t = -\nabla\cdot(\rho_t v_t)$ describes the evolution of a density where each particle flows along a vector ...
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How can I linearize the following equation (Bergman model)? [closed]

I have to linearize the following equation so that I can use the Laplace transform and get the transfer function for the system. The equation is: $$\frac{dG(t)}{dt}=-p_1 G(t)-p_2 X(t)G(t)+ ....$$ $p_1$...
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Generalized alignement index of classic Lorenz system?

I am reading about generalized alignment index (GALIs) as chaos indicator. However, I have been looking around for a while now to see an example of this applied on to the classic Lorenz attractor, but ...
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1 vote
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Soft question - Index theory in nonlinear dynamics vs Complex analysis

The video https://www.youtube.com/watch?v=wZvFKcQ_3Rc&t=8s mentioned something called the Index Theory. I can't find it on wikipedia. Where could I find more about the theory? Here index is just ...
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