All Questions
Tagged with nonlinear-dynamics nonlinear-system
135
questions
2
votes
0
answers
47
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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
3
votes
1
answer
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
0
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0
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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
1
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0
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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}...
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
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
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 ...
4
votes
1
answer
146
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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
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22
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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 ...
5
votes
1
answer
120
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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
1
vote
1
answer
73
views
General method for finding invariant subsapces of a nonlinear system
Suppose we are given a system:
$$\dot{x_{1}} = f_{1}(x_{1},...,x_{n})$$
$$...$$
$$\dot{x_{n}} = f_{n}(x_{1},...,x_{n})$$
And are interested in finding subspaces of the vector space that are invariant ...
- 402
0
votes
0
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94
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What should I prove to show the states lie within a compact set?
I'm trying to prove the local stability of a nonlinear system and got the following inequality.
$
\|x(t)\|\leq c_1\|x(t_0)\|\exp(-c_2(t-t_0))+c_3\epsilon_m\cdots
$(i)
where $c_1, c_2, c_3$ are ...
- 165
0
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0
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47
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Parameter Estimation of Dynamical System when Model is known
Im working on a nonlinear control based on Lyapunov theory and its working really well. I am able to implement it on a dynamical model of the system in simulink. However I think it has a really big ...
- 79
1
vote
0
answers
47
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Duffing equation with non-linearity factor greater than unity
I have been trying to solve the following non-linear equation taking help from the book regarding perturbative technique by H. Nayfeh (chapter -4)
$$\ddot{x}+\frac{x}{(1-x^2)^2}=0$$ where $x<1$ ...
- 344