All Questions
Tagged with nonlinear-dynamics ordinary-differential-equations
193
questions
1
vote
1
answer
62
views
Unsolvable characteristic system ODE as a part of PDE solution?
I'm trying to solve the following PDE:
$$F(x_1,x_2,u,p_1,p_2)=\text{ln}(x_2)p_1+x_2up_2-u=0 \ \ \ \ \ \ p_i=\partial_iu(x_1,x_2)$$
Where the initial conditions are:
$$\begin{cases}x_1(t)=t+1 \\x_2(t)=...
- 379
0
votes
1
answer
59
views
Orbit of vector field crosses transverse section in the same direction
Let $X\in\mathbf{C}^1(U,\mathbb{R}^2)$ a vector field on the open set $U\subset\mathbb{R}^2$. Let $D\subset\mathbb{R}$ open and $f:D\rightarrow U$ be a $\mathbf{C}^1$ map such that $\{f'(x),X_{f(x)}\...
- 625
1
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1
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46
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Exponential of nonlinear operator for a Cauchy problem
Does the exponential of a nonlinear operator solve the Cauchy problem for an ODE of say, this form
\begin{align*}
&\frac{dy}{dt}=f(t,y(t))\\
&y(0)=y_0
\end{align*}
so is this true?
\begin{...
- 320
2
votes
1
answer
37
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Stability of Hamiltonian system on degenerate critical point.
I'm trying to find information on the stability of the following ODE:
$$ x'' = x^4-x^2.$$
We know that it has a Hamiltonian $H(x,y) = \dfrac{y^2}{2} - (\dfrac{x^5}{5} - \dfrac{x^3}{3})$. The orbits ...
- 327
1
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0
answers
29
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Finding and classifying Hénon map bifurcations and periodic points
I am stumped on how to answer the following question:
Consider the Hénon map given by $$\textbf{H}(x, y) = (a-x^2+by, x)$$
Assume $0<b<1$. Classify the bifurcations that occur at $a = -\frac{1}{...
- 99
4
votes
0
answers
170
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Dynamics of a sliding cube on the $XY$ and $YZ$ planes
A cube with side length $a$, is initially placed with one vertex at the origin, and its faces parallel to the coordinate planes ($XY, XZ, YZ$) and totally lying in the first octant. Then its rotated ...
- 24.4k
3
votes
1
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124
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A question on the qualitative analysis of solution of a system of ODEs [closed]
Let $f:\mathbb{R}^2 \to \mathbb{R}^2$ be a non-zero smooth vector field satisfying $\text{div} f \ne 0.$ Which of the following are necessarily true for the ODE: $\dot{\mathbf{x}}=f(\mathbf{x})$?
(a) ...
- 779
2
votes
1
answer
63
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Classifying a second order non-linear ODE
I am currently dealing with the following ODE as a stationary, special case version of a PDE model derived from Kuramoto-Sivashinsky.
$$
y'' y' = ay
$$
Where $a$ is a real (constant) parameter.
I am ...
- 35
1
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0
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18
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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=...
- 11
1
vote
0
answers
59
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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}...
0
votes
1
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87
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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 ...
- 2,531
2
votes
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
0
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0
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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
0
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0
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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 ...
