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
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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
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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
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1
answer
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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
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0
answers
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
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answers
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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 $\alpha$, $\Omega$, ...
- 77
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0
answers
10
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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
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answers
21
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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
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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
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0
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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
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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
0
votes
1
answer
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
4
votes
1
answer
171
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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
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answers
45
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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
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37
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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
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answers
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 ...
