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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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Finding and classifying Hénon map bifurcations and periodic points

I am stumped on how to answer the following question: Consider the Hénon map given by $$\textbf{H}(x, y) = (a-x^2+by, x)$$ Assume $0<b<1$. Classify the bifurcations that occur at $a = -\frac{1}{...
JOlv's user avatar
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41 views

Dynamics of a sliding cube on the $XY$ and $YZ$ planes

A cube with side length $a$, is initially placed with one vertex at the origin, and its faces parallel to the coordinate planes ($XY, XZ, YZ$) and totally lying in the first octant. Then its rotated ...
c'est pas normale's user avatar
2 votes
0 answers
45 views

How to approximate any line segment within a circular region using the minimum number of connected rotating axes

This problem arises from my personal experience in developing a game mod. At that time, I wanted to create a drone system for vehicles, but due to the limitations of the game itself, I could only ...
S PLATEX's user avatar
1 vote
1 answer
18 views

Can probe trajectories to compute Lyapunov exponents get "stuck in more regular orbits" after rescaling?

I am computing Lyapunov exponents, and there is something that I do not understand about the data. The model has a regime for $\delta \approx 1$ (in some units) where it is fully chaotic, and the ...
Robin's user avatar
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Overdamped bead on rotating hoop

I have been trying to solve the following question but i am very unsure about my solution, can someone help me with it? Consider a bead of mass $m$ that slides along a circular rigid wire hoop of ...
Roozbeh Ranjbar's user avatar
3 votes
1 answer
106 views

A question on the qualitative analysis of solution of a system of ODEs [closed]

Let $f:\mathbb{R}^2 \to \mathbb{R}^2$ be a non-zero smooth vector field satisfying $\text{div} f \ne 0.$ Which of the following are necessarily true for the ODE: $\dot{\mathbf{x}}=f(\mathbf{x})$? (a) ...
MathRookie2204's user avatar
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22 views

Logistic map: bifurcation and domain of attraction

Let $f(x) = \mu x(1-x)$ be the logistic map, the question is divided into 3 parts: Part (1): what can you say about the domain of attraction of the 2-cycle in $3<\mu<1+\sqrt 6$? My attempt: let $...
vegetandy's user avatar
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2 votes
1 answer
63 views

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 ...
Vasil's user avatar
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2 votes
0 answers
48 views

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 ...
Noumeno's user avatar
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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!
confusion's user avatar
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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 $n \times n$...
MIKE PAPADAKIS's user avatar
1 vote
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17 views

Which is the Theorem to demonstrate positivity in a system of nonlinear ODEs?

Let $X'(t) = f(X), X(0) = X_0$ be a system of nonlinear ODEs with a positive initial condition, and f is Lipschitz continuous. In a forum, I read that whenever $f_i(X) \ge 0$ if $X_i = 0,$ for all $i=...
Jesús's user avatar
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Mean, Variance and Correlation Function of a quadratic SDE

I am struggling with the following nonlinear SDE: $ ds=dt(-\Omega s^2(t)+\alpha s(t)+\beta) + d\xi(t)(\gamma (1-s(t))) $ $ d\xi = dt(-\frac{1}{\tau} \xi(t)) + \sigma dW(t) $ Where $\alpha$, $\Omega$, ...
duodenum's user avatar
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10 views

Example 1.2 Nonlinear Control Khalil

$f( x) =\begin{bmatrix} x_{2}\\ -sat( x_{1} +x_{2}) \end{bmatrix}$ is not continuously differentiable on $R^2$. Using the fact that the saturdation function sat(.) satisfies $|sat(\eta)-sat{\xi}|$, we ...
SS1's user avatar
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14 views

Complex valued Hamilton Jacobi equation

Let $g_{ij}(t,x)$ be a metric tensor with dependence on t,x. Consider $$\partial_t u(t,x) = i\sqrt{\sum_{i,j} g_{ij}\partial_iu\partial_ju},u(0,x)=u_0(x).$$ Where $u(t,x):\mathbb{R}\times\mathbb{R}^n\...
xinggu's user avatar
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9 votes
2 answers
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 ...
krishnab's user avatar
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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}...
Andris Erglis's user avatar
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30 views

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 ...
vivvv's user avatar
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0 votes
1 answer
82 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 ...
krishnab's user avatar
  • 2,491
4 votes
1 answer
163 views

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 ...
LCG's user avatar
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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 ...
Leszek's user avatar
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0 answers
36 views

Nonlinear Dynamics and Chaos Strogatz Question 4.4.3

Over dampened Pendulum System: $$ mL^{2}\ddot{\theta } +b\dot{\theta } +mgL\sin \theta =\Gamma $$ First order approximation: $$ b\dot{𝜃}+mgL\sin{}𝜃=Γ $$ Nondimensionalize, diving through by mgL: $$ ...
SS1's user avatar
  • 77
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0 answers
30 views

Any common reference for linear, TIME-VARYING systems?

This isn't exactly a math question (apologies!), but it could prevent many potential misunderstandings I might otherwise encounter in the near future (and also prevent many dumb questions I will post ...
lostintimespace's user avatar
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0 answers
12 views

Finding a topological conjugacy of one-parameter quadratic families

Let $f_c : z \rightarrow z^2 +c$ and $Q_a: x \rightarrow ax(1-x)$, I have to show that for $c \in [-2, \frac{1}{4}]$ there is an $a\in[1,4]$ such that $f_c$ is conjugate to $Q_a$ Unfortunately, I'm ...
variableXYZ's user avatar
  • 1,073
2 votes
1 answer
61 views

Every ergodic invariant measure of one dimensional dynamical system is a Dirac measure

Recently, I have learned some theorems about attractors and invariant measures. In the book I am reading, there is a theorem presented without its proof. I am interested in how to prove it. Recall ...
R-CH2OH's user avatar
  • 195
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0 answers
43 views

If we perturb an ODE, with the same starting conditions can we show that they converge together over time or at least do not diverge?

Suppose we have two ODEs: $\dot{x}(t) = f(x(t),t)$ $\dot{y}(t) = f(y(t),t) + g(y(t),t)$ If we have identical starting conditions $x(0) = y(0)$, we see $$y(t) - x(t) = \int_0^t \left[ f(y(t),t)-f(x(...
travelingbones's user avatar
6 votes
1 answer
140 views

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.,...
dang's user avatar
  • 105
0 votes
0 answers
34 views

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 ...
Brayan Guerra's user avatar
2 votes
1 answer
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_{...
matteogost's user avatar
0 votes
0 answers
69 views

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 ...
SR9054505's user avatar
2 votes
0 answers
71 views

Does the Hamiltonian system have unbound solutions?

I want to know if it is possible to determine if the following Hamiltonian system has unbound solutions. Let us consider the Hamiltonian function $$ H(x,y,p_x,p_y) = \frac{1}{2}(p_x^2+p_y^2) + \frac{x^...
alejandro's user avatar
  • 123
2 votes
0 answers
84 views

How to get a Filippov solution?

Recently, I read a book ISSN 2195-9862 about the Filippov theory. There is a differential inclusion $$\dot{x}\in F(x)=\begin{cases} -1&x>0\\ [-1,1]&x=0\\ 1&x<0 \end{cases}\\ x(t_0)=...
Liu C's user avatar
  • 21
2 votes
0 answers
46 views

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 ...
Yonatan's user avatar
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1 vote
1 answer
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 ...
Olayo's user avatar
  • 67
4 votes
1 answer
141 views

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) &...
Trb2's user avatar
  • 380
0 votes
0 answers
19 views

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 ...
Abheek Ghosh's user avatar
0 votes
0 answers
47 views

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 ...
Yonatan's user avatar
  • 35
0 votes
0 answers
20 views

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 ...
Axel Wang's user avatar
3 votes
0 answers
73 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 ...
duodenum's user avatar
1 vote
0 answers
21 views

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 ...
Juno Kim's user avatar
  • 610
2 votes
0 answers
47 views

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: $$\frac{dG(t)}{dt}=-p_1 G(t)-p_2 X(t)G(t)+ ....$$ $p_1$...
ChMic's user avatar
  • 29
0 votes
0 answers
7 views

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 ...
Axel Wang's user avatar
5 votes
1 answer
119 views

Why is this approximate solution correct?

Consider the following differential equation $$ y''=-y + \alpha y |y|^2, $$ where $y=y(x)$ is complex in general and $\alpha$ is a real constant such that the second term is small compared to $y$ ($||^...
user655870's user avatar
1 vote
1 answer
61 views

General method for finding invariant subsapces of a nonlinear system

Suppose we are given a system: $$\dot{x_{1}} = f_{1}(x_{1},...,x_{n})$$ $$...$$ $$\dot{x_{n}} = f_{n}(x_{1},...,x_{n})$$ And are interested in finding subspaces of the vector space that are invariant ...
Mani's user avatar
  • 402
2 votes
0 answers
65 views

Example of a buried Julia component of a transcendental meromorphic function.

We know examples of buried Julia components (Definition: A Julia component is called buried if it is not contained in the boundary of any Fatou component) for rational functions. In 1998, McMullen ...
Factorial_zero's user avatar
0 votes
0 answers
70 views

Mathematical theory of plasma

I am working on a heavily mathematically project about plasma. In particular, I want to find references that treat the problem from microscopic models that include relativistic and magnetic effects (...
The N's user avatar
  • 113
0 votes
0 answers
16 views

Topological conjugacy of the logistic map at different parameter values

I am wondering whether the dynamical systems generated by the discrete 1 dimensional map $g(x;p) = px(1-x)$ (the logistic map) at different values of $p$ are topologically conjugate. Of course, this ...
its_all_a_DS's user avatar
0 votes
0 answers
93 views

What should I prove to show the states lie within a compact set?

I'm trying to prove the local stability of a nonlinear system and got the following inequality. $ \|x(t)\|\leq c_1\|x(t_0)\|\exp(-c_2(t-t_0))+c_3\epsilon_m\cdots $(i) where $c_1, c_2, c_3$ are ...
SpaceTAKA's user avatar
  • 165
1 vote
0 answers
53 views

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 ...
HIH's user avatar
  • 361
0 votes
0 answers
47 views

nonlinear odes: stabilizing terms in a subcritical pitchfork bifurcation

I am reading through Strogatz's book on nonlinear odes and dynamical systems. One thing that is a little confusing is his description of stabilizing higher order terms to control the dynamics of a ...
krishnab's user avatar
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