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.
486
questions
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12
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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 ...
- 1,073
2
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1
answer
70
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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 ...
- 351
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45
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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
$$y(t) - x(t) = \int_0^t \left[ f(y(t),t)-f(x(...
- 286
6
votes
1
answer
141
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Are there general solutions to quadratic, 2D, continuous, time-invariant dynamical systems?
I am a bit new to dynamical systems and don't know my way around terminology, so have had a hard time answering this for myself.
I know the basics of theory for 2D linear, time-invariant systems, i.e.,...
- 105
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46
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How to calculate the monodromy matrix for a system of nonautonomous nonlinear differential equations?
I am interested in analyzing the stability of the periodic orbits resulting from the Van der Pol system periodically perturbed by a time-dependent external forcing. Mathematically it would be the ...
2
votes
1
answer
67
views
Final value of a recursion
Problem
Given $p_1, \sigma > 0$, consider the following recursion
\begin{equation*}
p_{i}=(1-L_i)p_{i-1} \qquad i=2,\dots,k
\end{equation*}
where
\begin{equation*}
L_i \triangleq \frac{p_{i-1}}{p_{...
- 729
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76
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Singularity of a non- linear second order ODE
I have the encountered a singularity in the equation below .
$$
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,
$$
with ...
2
votes
0
answers
71
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Does the Hamiltonian system have unbound solutions?
I want to know if it is possible to determine if the following Hamiltonian system has unbound solutions. Let us consider the Hamiltonian function
$$ H(x,y,p_x,p_y) = \frac{1}{2}(p_x^2+p_y^2) + \frac{x^...
- 123
2
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115
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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)=...
- 21
2
votes
0
answers
60
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 ...
- 35
1
vote
1
answer
109
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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 ...
- 87
4
votes
1
answer
146
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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) &...
- 380
0
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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 ...
- 11
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52
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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 ...
- 35
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0
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22
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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 ...
