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
-1
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0
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30
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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
2
votes
0
answers
47
views
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
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:
$$\begin{cases}x_1(t)=t+1 \\x_2(t)=...
- 379
0
votes
1
answer
59
views
Orbit of vector field crosses transverse section in the same direction
Let $X\in\mathbf{C}^1(U,\mathbb{R}^2)$ a vector field on the open set $U\subset\mathbb{R}^2$. Let $D\subset\mathbb{R}$ open and $f:D\rightarrow U$ be a $\mathbf{C}^1$ map such that $\{f'(x),X_{f(x)}\...
- 625
1
vote
1
answer
46
views
Exponential of nonlinear operator for a Cauchy problem
Does the exponential of a nonlinear operator solve the Cauchy problem for an ODE of say, this form
\begin{align*}
&\frac{dy}{dt}=f(t,y(t))\\
&y(0)=y_0
\end{align*}
so is this true?
\begin{...
- 320
1
vote
0
answers
35
views
How to accurately average a function with a nonlinear response?
I am a physics PhD student working in optics and I have a bit of a weird problem that I am trying to sort out and I'm hoping you math folks can help me with.
Without boring you with the experimental ...
2
votes
1
answer
37
views
Stability of Hamiltonian system on degenerate critical point.
I'm trying to find information on the stability of the following ODE:
$$ x'' = x^4-x^2.$$
We know that it has a Hamiltonian $H(x,y) = \dfrac{y^2}{2} - (\dfrac{x^5}{5} - \dfrac{x^3}{3})$. The orbits ...
- 327
1
vote
0
answers
29
views
Finding and classifying Hénon map bifurcations and periodic points
I am stumped on how to answer the following question:
Consider the Hénon map given by $$\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
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 ...
- 24.4k
2
votes
0
answers
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
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 ...
- 31
1
vote
0
answers
34
views
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
124
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) ...
- 779
0
votes
0
answers
25
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 $...
- 305
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 ...
- 35
2
votes
0
answers
50
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 ...
- 363
0
votes
0
answers
13
views
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!
- 53
0
votes
1
answer
59
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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$...
1
vote
0
answers
18
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=...
- 11
1
vote
0
answers
27
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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$, ...
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0
votes
0
answers
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 ...
- 79
0
votes
0
answers
21
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\...
- 1
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 ...
- 2,531
1
vote
0
answers
59
views
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
0
answers
31
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 ...
- 1
0
votes
1
answer
87
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 ...
- 2,531
4
votes
1
answer
171
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 ...
- 51
0
votes
0
answers
45
views
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 ...
- 91
0
votes
0
answers
37
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
$$
Nondimensionalize, diving through by mgL:
$$
...
- 79
0
votes
0
answers
31
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 ...
0
votes
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 ...
- 1,073
2
votes
1
answer
70
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 ...
- 351
0
votes
0
answers
45
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
$$y(t) - x(t) = \int_0^t \left[ f(y(t),t)-f(x(...
- 286
6
votes
1
answer
141
views
Are there general solutions to quadratic, 2D, continuous, time-invariant dynamical systems?
I am a bit new to dynamical systems and don't know my way around terminology, so have had a hard time answering this for myself.
I know the basics of theory for 2D linear, time-invariant systems, i.e.,...
- 105
0
votes
0
answers
46
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How to calculate the monodromy matrix for a system of nonautonomous nonlinear differential equations?
I am interested in analyzing the stability of the periodic orbits resulting from the Van der Pol system periodically perturbed by a time-dependent external forcing. Mathematically it would be the ...
2
votes
1
answer
67
views
Final value of a recursion
Problem
Given $p_1, \sigma > 0$, consider the following recursion
\begin{equation*}
p_{i}=(1-L_i)p_{i-1} \qquad i=2,\dots,k
\end{equation*}
where
\begin{equation*}
L_i \triangleq \frac{p_{i-1}}{p_{...
- 729
0
votes
0
answers
76
views
Singularity of a non- linear second order ODE
I have the encountered a singularity in the equation below .
$$
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,
$$
with ...
2
votes
0
answers
71
views
Does the Hamiltonian system have unbound solutions?
I want to know if it is possible to determine if the following Hamiltonian system has unbound solutions. Let us consider the Hamiltonian function
$$ H(x,y,p_x,p_y) = \frac{1}{2}(p_x^2+p_y^2) + \frac{x^...
- 123
2
votes
0
answers
115
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)=...
- 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
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 ...
- 87
4
votes
1
answer
146
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) &...
- 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 ...
- 11
0
votes
0
answers
52
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 ...
- 35
0
votes
0
answers
22
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 ...
3
votes
0
answers
84
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 ...
- 77
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 ...
- 610
2
votes
0
answers
51
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)+ ....$$
$p_1$...
- 29
0
votes
0
answers
7
views
Generalized alignement index of classic Lorenz system?
I am reading about generalized alignment index (GALIs) as chaos indicator. However, I have been looking around for a while now to see an example of this applied on to the classic Lorenz attractor, but ...
5
votes
1
answer
120
views
Why is this approximate solution correct?
Consider the following differential equation
$$ 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 $y$ ($||^...
- 331