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8 votes
2 answers
1k views

What is the current status on methods to find limit cycles?

What are the current best methods to show analytically the existence of a limit cycle in a $n$-dimensional system of the form: $$ \frac{\mathrm{d}}{\mathrm{d} t} \vec{x}(t)=\vec{f}(\vec{x}) $$ Where $...
Matt's user avatar
  • 117
7 votes
1 answer
930 views

(In)stability of a two-dimensional dynamical system

Consider the following system of coupled differential equations \begin{eqnarray*} \dot{x}_1(t) & = & -x_1(t) - \cos(\omega t)x_1(t) + \cos(\omega t)x_2(t), \ x_1(0)\in\mathbb{R},\\ \dot{x}_2(t)...
Ludwig's user avatar
  • 2,712
5 votes
1 answer
211 views

Stability of ODEs with polynomial nonlinearity

Consider the following ODE system: $$ x′=f(x)\iff \begin{pmatrix} x_1^\prime \\ \vdots\\ x_k^\prime\\ \vdots\\ x_n ^\prime \end{pmatrix} = \begin{pmatrix} f_1(x) \\ \vdots\\ f_k(x)\\ \vdots\\ f_n(x) \...
Zhang Yuhan's user avatar
5 votes
0 answers
281 views

Basin of attraction of gradient flow

Suppose we have a compact Riemannian manifold $(M,g)$, and a Morse function $f : M \rightarrow \mathbb{R}$. Suppose we consider the gradient flow generated by this function, i.e. $$\dot{x_t} = - \...
Rikimaru's user avatar
  • 151
3 votes
2 answers
428 views

Nonlinear ODE system: stability

I've got this 4x4 system that should model the wine fermentation process. All the $\mu, K_N, k_d$ etc are positive constants. Of course I have no idea of how to solve it. But at least I would like to ...
7iat's user avatar
  • 31
3 votes
1 answer
202 views

Stability of nonsmooth, Lipschitz continuous, autonomous system of differential equations

Consider the following autonomous system of differential equations: $$\frac{\mathrm d\mathbf x}{\mathrm dt} = \mathbf v(\mathbf x)$$ where $\mathbf x, \mathbf v \in \mathbb R^n$. Assume that $\...
valle's user avatar
  • 884
3 votes
1 answer
252 views

Stability analysis of equilibrium point of non-linear ODE system with Jacobian going to infinity

Let's say I have a nonlinear system of ODEs, where one of equations looks like: $$ \frac{dX_i}{dt} = a_1X_0+\dotsb+a_j\frac{X_j^{0.5}}{X_j^{0.5}+b_j^{0.5}}+\dotsb. $$ And equilibrium point is 0. I ...
Omega's user avatar
  • 31
3 votes
0 answers
50 views

Stability of indefinitely damped mechanical system with diagonal stiffness

I'm trying to find conditions for the asymptotic stability of the following linear system, \begin{equation} \mathbf{I \ddot{x}} + \mathbf{B \dot{x}} + \mathbf{K x} = 0 \end{equation} given the ...
Shivang Rawat's user avatar
2 votes
1 answer
239 views

Seeking a Lyapunov function for a SIR model with immunity loss

We add the immunity loss to the SIR model and obtain the following autonomous system. $$ \begin{align} s' &= -is+\alpha r \\ i' &= i s - \gamma i\\ r' &= \gamma i-\alpha r \end{align} \...
Hans's user avatar
  • 2,239
2 votes
1 answer
134 views

On local attractivity of a coupled non-linear differential equation

Consider a dynamical system described by the following coupled non-linear differential equation \begin{align} \dot{x}_1(t) &= v + a_{12}\sin(x_2(t)-x_1(t)) + a_{13}\sin(x_3(t)-x_1(t))\\ \dot{x}_2(...
Ludwig's user avatar
  • 2,712
2 votes
1 answer
65 views

Boundedness of particle motion with time-varying force

Consider the differential equation $$ m \ddot{x} + k \dot{x} = - W_t x $$ where $m$ and $k$ are nonnegative. $x_t \in \mathbb{R}^n$ $W_t$ is a matrix that satisfies $$ \alpha I \succeq W_t \...
Michael S.'s user avatar
2 votes
0 answers
153 views

Stability test for LTV systems by differential Lyapunov inequalities

Consider a linear time-varying system: \begin{equation} \dot x(t) = A(t) x(t), \tag{$*$} \end{equation} where $A(t)$ is a time-varying block matrix defined as $$ A(t) = \begin{bmatrix} 0 & I\\ -\...
Evan's user avatar
  • 55
1 vote
0 answers
53 views

Basin of attraction comparative statics* using local energy functions?

Let $\dot{\boldsymbol{x}}=\boldsymbol{F}(\boldsymbol{x};p)$ be an autonomous dynamical system defined on $[a,b]^n$ ($-\infty<a<B<\infty)$; $p\in\mathbb{R}$ is some fixed parameter. Suppose ...
cluelessmathematician's user avatar
1 vote
1 answer
151 views

Analytically characterizing basins of attraction boundaries and sizes

While I understand that doing the above is not possible in general, I would like to know more about how to proceed when it is possible. That is, what are the common methods people use to analytically ...
cluelessmathematician's user avatar
1 vote
0 answers
62 views

Lyapunov theory in coupled nonlinear dynamic system with input

Suppose I have the following nonlinear coupled dynamic system \begin{align*} &\dot{x}_1 = f_1(x_1,x_2)\\ &\dot{x}_2 = f_2(x_2) + u \end{align*}where $x_1\in \mathbb{R}^{n_1}$, $x_2\in \mathbb{...
sleeve chen's user avatar
1 vote
0 answers
44 views

Sufficient conditions for the continuity of an improper integral concerning the finite-time stability of a dynamical system

Consider the initial value problem \begin{equation}\label{fainait ve} \dot{\boldsymbol{x}}(t) = \boldsymbol{f}(\boldsymbol{x}(t)), \;\; t \geq 0, \; \;\boldsymbol{f}(\boldsymbol{0}_n) = \boldsymbol{0}...
Johannes's user avatar
1 vote
0 answers
276 views

Stability when linearization fails

The dynamics of the $j$th system: \begin{equation} \begin{split} \dot{\overline r}_j &= h (\overline r_j) \,\, - \varepsilon \omega_{\mathrm{sw}}R_{\mathrm{Th}}\sqrt{\xi^2+\chi^2}\sum^N_{k=1} \...
Mohit's user avatar
  • 33
0 votes
1 answer
285 views

Quadratic stability of linear time varying system

(This question was originally asked at Math.SE, where it didn't receive any answers.) Consider the linear time-varying system $$ \dot{x} = A(t) x, $$ where $x \in \mathbb{R}^n$ and $A: [0,+\infty) \...
Tadashi's user avatar
  • 1,590
0 votes
1 answer
100 views

Stabilize the vector field of $y' = f (y) - \gamma H^T(HH^T)^{-1}h( y ) $ of ODE $y' = f(y)$

This question has been asked here but there is no answer: https://math.stackexchange.com/questions/1585400/stabilize-the-vector-field-of-y-f-y-hthht-1h-y-of-ode-y Consider autonomous ODE $y' = ...
David Li's user avatar
  • 103
0 votes
0 answers
320 views

Prove that origin is globally exponentially stable with Lyapunov Indirect Method

I'm wondering, if we have a nonlinear system governed by $\dot{x} = Ax + g(x)$ where $||g(x)|| \leq \gamma ||x||^2$ and A is Hurwitz how can we show that the origin is globally exponentially stable?...
Aerandir's user avatar