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
11
votes
1
answer
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What is the most general Carathéodory-type global existence theorem?
I am looking for a general theorem that guarantees the existence of a global solution for an ODE system in $\mathbb{R}^n$
$$
\begin{equation}
\left\{ \begin{aligned}
x'(t) &= f(t, x(t)), \qquad t \...
- 761
9
votes
2
answers
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views
What is meant when mathematicians or engineers say we cannot solve nonlinear systems?
I was watching a video on "system identification" in control theory, in which the creator says that we don't have solutions to nonlinear systems. And I have heard this many times in many ...
- 2,531
8
votes
1
answer
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views
Stability analysis of the dynamical system $\ddot{x}+b\dot{x}+K x-\|\dot{x}\| \frac{x-x_i}{\|x-x_i\|^2}=0$ .
Consider the dynamical system described as:
$$\ddot{z}+b\dot{z}+ K z-\|\dot{z}\| \frac{z-z_i}{\|z-z_i\|^3}=0$$
where $z=[x \ \ y]^T$, $K$ is a positive definite matrix and $b \in \mathbb{R}$, I made ...
- 467
8
votes
0
answers
320
views
Extension of Burgers' equation
I recently encountered a viscous Burgers' equation type PDE, but with the addition of a derivative-squared nonlinear term (in dimensionless form):
$u_t - u_{xx} + uu_x - u_x^2 = 0\,,$
where the ...
- 71
7
votes
1
answer
183
views
Generalisation of Index of a curve to higher dimensions
Im studying Non Linear Dynamics and Chaos from Strogatz's textbook. In the sixth chapter, while talking about non linear flows in 2 dimensions he introduces the index of a curve in a vector field and ...
- 3,560
6
votes
1
answer
141
views
Are there general solutions to quadratic, 2D, continuous, time-invariant dynamical systems?
I am a bit new to dynamical systems and don't know my way around terminology, so have had a hard time answering this for myself.
I know the basics of theory for 2D linear, time-invariant systems, i.e.,...
- 105
6
votes
1
answer
89
views
An ODE confusion
I was thinking about a ODE problem recently when I was reading about dynamical system. In school we used to solve the ODE problem $\frac{dx}{dt}=\sqrt{1-x^2}, x=0, t=0$ as $x=\sin(t),$ which will have ...
- 9,098
6
votes
1
answer
205
views
How does this expression follow algebraically from the last one?
I was reading this paper:
Global stability for an HIV/AIDS epidemic model with different latent stages and treatment
Everything is understood apart from on page 7 of the pdf (page 1486 in the document)...
user644376
6
votes
0
answers
155
views
Nature of ODE $\dot x=x^2-\frac{t^2}{1+t^2}$
Discuss the equation $\dot{x}=x^2-\frac{t^2}{1+t^2}$.
Make a numerical analysis.
Show that there is a unique solution which asymptotically approaches the line $x=1$.
Show that all solutions below ...
- 9,098
6
votes
0
answers
129
views
Estimate on derivative of ODE solution with respect to parameters
Consider the ODE
$$
u'(t) = f(t,u,p), \qquad u(0) = v
$$
where $p$ is a control parameter, and let $u(t;v,p)$ denote the solution to the problem above for fixed $v$ and $p$.
It is apparently "...
5
votes
1
answer
65
views
Inequality of a linearized function
The following inequality is used in one of the steps for the proof of the centre manifold theorem. Consider the function $$f(x)=Ax+\tilde{f}(x)$$
where $x\in \mathbb{R}^n$, $f:\mathbb{R}^n \rightarrow ...
- 153
5
votes
1
answer
812
views
Does Asymptotic Stability Imply the Existence of a Lyapunov Function for a Nonlinear System?
For a linear time-invariant system $\dot x = Ax,$ the inverse Lyapunov theorem asserts that if the origin is asymptotically stable, then a Lyapunov function in the form $V(x) = x^\top P x$ for some ...
- 394
5
votes
1
answer
111
views
How to prove this theorem for the number of components of a filled julia sets?
If one finite critical point of $f(z)$ escapes to infinity by iterating, then the filled-in Julia set of $f(z)$ consists of infinitely many components.
How to prove this ?
I must admit I heard this in ...
- 16.4k
5
votes
1
answer
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views
$f(n) = \frac{n^2 + n + 4}{2}, g(f(n)) = f(g(n))$ such that $g(n)$ is an integer.
Let $n$ be a strict positive integer.
Lets define an integer sequence $f(n)$ :
$$f(n) = \frac{n^2 + n + 4}{2}$$
so
$$f(1) = 3$$
$$f := {3,5,8,12,17,23,30,38,47,...}$$
$$f(17) = 155$$
etc
Notice
$$3+...
- 16.4k
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