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
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Generate trajectory between 2 points to achieve a desired momentum
I have 2 points and I need to find a path between them to maximize momentum.
You can consider this as a trajectory of a Racquet hitting a tennis ball.
Current_Trajectory
In the image above, the ...
- 1
3
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1
answer
174
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Poincaré-Bendixson applied to region enclosed by homoclinic orbit
Let $p$ be a saddle point of the planar ODE $x' = f(x)$ with $f$ smooth. Suppose $\gamma$ is a homoclinic orbit starting and ending in $p$.
Define $\Gamma := \gamma \cup p$ and let $\mathcal{U}$ be ...
- 211k
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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{...
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Which nonlinear PDEs can be converted into linear PDEs?
In Section 4.4 of Partial Differential Equations by Evans, the author describes several techniques for converting certain nonlinear equations into linear equations. First, the author introduces the ...
- 356
1
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1
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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:
$$\begin{cases}x_1(t)=t+1 \\x_2(t)=...
- 66.8k
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1
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59
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Orbit of vector field crosses transverse section in the same direction
Let $X\in\mathbf{C}^1(U,\mathbb{R}^2)$ a vector field on the open set $U\subset\mathbb{R}^2$. Let $D\subset\mathbb{R}$ open and $f:D\rightarrow U$ be a $\mathbf{C}^1$ map such that $\{f'(x),X_{f(x)}\...
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Dynamics of a sliding cube on the $XY$ and $YZ$ planes
A cube with side length $a$, is initially placed with one vertex at the origin, and its faces parallel to the coordinate planes ($XY, XZ, YZ$) and totally lying in the first octant. Then its rotated ...
- 24.4k
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35
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How to accurately average a function with a nonlinear response?
I am a physics PhD student working in optics and I have a bit of a weird problem that I am trying to sort out and I'm hoping you math folks can help me with.
Without boring you with the experimental ...
2
votes
1
answer
37
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Stability of Hamiltonian system on degenerate critical point.
I'm trying to find information on the stability of the following ODE:
$$ x'' = x^4-x^2.$$
We know that it has a Hamiltonian $H(x,y) = \dfrac{y^2}{2} - (\dfrac{x^5}{5} - \dfrac{x^3}{3})$. The orbits ...
- 15.5k
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Finding and classifying Hénon map bifurcations and periodic points
I am stumped on how to answer the following question:
Consider the Hénon map given by $$\textbf{H}(x, y) = (a-x^2+by, x)$$
Assume $0<b<1$. Classify the bifurcations that occur at $a = -\frac{1}{...
- 99
1
vote
1
answer
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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 ...
- 31
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47
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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 ...
- 21
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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 ...
3
votes
1
answer
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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) ...
- 211k
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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 $...
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