All Questions
Tagged with nonlinear-dynamics dynamical-systems
166
questions
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Orbit of vector field crosses transverse section in the same direction
Let $X\in\mathbf{C}^1(U,\mathbb{R}^2)$ a vector field on the open set $U\subset\mathbb{R}^2$. Let $D\subset\mathbb{R}$ open and $f:D\rightarrow U$ be a $\mathbf{C}^1$ map such that $\{f'(x),X_{f(x)}\...
- 625
2
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1
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37
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Stability of Hamiltonian system on degenerate critical point.
I'm trying to find information on the stability of the following ODE:
$$ x'' = x^4-x^2.$$
We know that it has a Hamiltonian $H(x,y) = \dfrac{y^2}{2} - (\dfrac{x^5}{5} - \dfrac{x^3}{3})$. The orbits ...
- 327
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29
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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}{...
- 99
1
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34
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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 ...
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25
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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 $...
- 305
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31
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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 ...
- 1
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31
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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 ...
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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
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1
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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
$$y(t) - x(t) = \int_0^t \left[ f(y(t),t)-f(x(...
- 286
6
votes
1
answer
141
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Are there general solutions to quadratic, 2D, continuous, time-invariant dynamical systems?
I am a bit new to dynamical systems and don't know my way around terminology, so have had a hard time answering this for myself.
I know the basics of theory for 2D linear, time-invariant systems, i.e.,...
- 105
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46
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How to calculate the monodromy matrix for a system of nonautonomous nonlinear differential equations?
I am interested in analyzing the stability of the periodic orbits resulting from the Van der Pol system periodically perturbed by a time-dependent external forcing. Mathematically it would be the ...
2
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71
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Does the Hamiltonian system have unbound solutions?
I want to know if it is possible to determine if the following Hamiltonian system has unbound solutions. Let us consider the Hamiltonian function
$$ H(x,y,p_x,p_y) = \frac{1}{2}(p_x^2+p_y^2) + \frac{x^...
- 123
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60
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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 ...
- 35
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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 ...
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