Questions tagged [limit-cycles]
A limit cycle is a closed trajectory in state space such that at least one other trajectory spirals into it or spirals out of it
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Generalised Dulac Theorem
Can someone explain the proof of the theorem, their are several thing i dont understand espacially how u obtain the fact , that C lays in $\Gamma$
If we change the region
R (simply connected Region) ...
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Hamilton paths skipping some vertex relations
I have been developing a Python code to compute all the Hamilton cycles for a system but excluding those that have a maximum distance d between vertices. Hence, for d=1 i would only have a single ...
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Homoclinic orbit vs limit cycle
In dynamical systems, what is the distinction between a homoclinic orbit and a limit cycle? It seems to me like a homoclinic orbit is effectively just a limit cycle with a particular fixed point along ...
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Can someone help me with good oscillation pictures corresponding to stable and unstable limit cycles?
I have seen a lot of pictures of stable and unstable limit cycles in 2D planes using nullclines and trajectories. I am trying to understand them using oscillations of a particle nearby. If you give me ...
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Crossing the limit cycle in a DDE
I have the following delay differential equation
$$
\begin{align}
\frac{dx_1(t)}{dt} &= \frac{1}{1 + \left(\frac{x_2(t-\tau)}{p_0}\right)^n} - \mu_m \cdot x_1(t)\\
\frac{dx_2(t)}{dt} &=...
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3 species Lotka–Volterra model. Limit cycle
Good day, I have 3 species Lotka–Volterra model. My goal is to determine if there is a limit cycle in the system
$$
\left\{
\begin{array}{l}
\frac{d c}{d t}=r_c c(1-c)-\frac{c h}{c+\theta_1} \\
\frac{...
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1
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Limit cycles of the system
$$\dot x=y+x[\mu-(x^2+y^2-1)^2]$$
$$\dot y=-x+y[\mu-(x^2+y^2-1)^2]$$
I used the polar coordinates substitution :
$$x=r\cos\theta$$
$$y=r\sin\theta$$
and via the expressions :
$$r\dot r=x\dot x+y\dot y$...
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Concept of $\alpha$ and $\omega$ - limit points
It is given that a point $(x^*,y^*)$ is said to be $\omega$- limit point ($\alpha$ -limit point) of the trajectory of a system if there exists a sequence of times $𝑡_𝑛 \to \infty (𝑡_𝑛 \to -\infty)$...
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Generalizing an application of the Poincaré-Bendixson theorem
I was looking for applications of the Poincaré-Bendixson theorem and on this site I have found several examples almost all similar to this post. So I tried to make a quite natural generalization
$$
\...
- 1,592
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Prove system has periodic solutions but no limit cycle
I have the following system
\begin{align*}
\dot{x} &= -y+xy,\\
\dot{y} &= x+\frac{1}{2}(x^2-y^2)\\
\end{align*}
I have to prove that this system has periodic solutions ...
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Limit cycle of ODE system
I have problems with the following exercise. Prove that the system
\begin{align*}
\dot{x} &= x(x^2+y^2-2x-3) - y\\
\dot{y} &= y(x^2+y^2-2x-3) + x
\end{align*}
has a cycle limit.
My ...
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0
answers
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Ruling out limit cycles in 2 dimensions
Let $f:[0,1]^2 \rightarrow \mathbb{R}^2,$ where $f_1(x,y) = g(y)-x$ and $f_2(x,y) = g(x)-y.$ Here $g(\cdot)$ is a strictly decreasing polynomial function such that $g(0)=1$ and $g(1)=0.$ I am ...
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Looking for a generalization of Lienard's theorem
Let $g:\mathbb R\to\mathbb R$ be a $C^1-$function such that
$G(u)=\displaystyle\int_0^ug(r)dr$ is odd
$\displaystyle\lim_{u\to+\infty}G(u)=+\infty$ and there is a $\beta>0$ such that if $u>\...
- 1,592
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Is there more than one way to derive an energy function from differential equations?
I don't know how this energy function (screenshot below) comes from the oscillator equation. I know you can get it from $E =\frac{{\dot x}^2}{2} - \int \ddot x (x)dx$, which is conservative (meaning $...
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Trapping Region for ODE System.
I am working on the following problem, given the system of two differential equations
$x′=2x+y−2x^3−3xy^2,$
$y′=−2x+4y−4y^3−2x^2y,$
So far, I have tackled similar problems by trying to find a trapping ...
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Show that the system has at least one limit cycle
$$x' = -y + x(1-2x^2 - 3y^2)$$
$$y' = x + y(1-2x^2 - 3y^2)$$
I've started by converting to polar coordinates
$$x=rcos\theta \quad y = rsin\theta, \quad rr'=xx'+yy'$$
This gives me
$$r' = 1-2r^2cos^2\...
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Proving a dynamical system has a limit cycle
I am trying to prove that the following dynamical system has a limit cycle by rewriting it in polar coordinates
$\dot{x} = x-y-x^2(x+2y)-xy^2$
$\dot{y}= x+y+x^2(x-y)-y^2(x+y)$
Using the identities
$r\...
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1
answer
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Limit cycle within $\frac{1}{4}<r<1$
Show that the system
\begin{align}
x'&=-y+x(1-2x^2-3y^2)\nonumber\\
y'&=x+y(1-2x^2-3y^2)\nonumber
\end{align}
has a limit cycle in $\frac{1}{4}<r<1$.
Here's what I've done so far:...
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Limit cycles of the system $ \dot{r}=\sin\pi r, \dot{\theta}=\cos\pi r. $
In polar coordinates, consider the system
$$
\dot{r}=\sin\pi r,\quad \dot{\theta}=\cos\pi r.
$$
Prove that the system has two limit cycles with one lying interior to the other and with no equilibrium ...
- 1,335
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Radius of limit cycle for van der Pol oscillator in the limit of $\varepsilon\ll1$ using Green's Theorem
We want to determine the radius of the nearly circular limit cycle of the van der Pol oscillator $$\ddot x + \varepsilon \dot x \left( x^2 - 1 \right) + x = 0$$ in the limit $\varepsilon\ll1$. Assume ...
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proof of a theorem about existence of stable limit cycles
Consider the nonlinear Kolmogorov system $$\frac{dx}{dt}=xf(x,y) \\
\frac{dy}{dt}=yg(x,y)$$ where $f,g$ are continuously differentiable
functions in $\mathbb{R}^2$. Then prove that the above system ...
- 2,028
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Existence of limit cycle inside annulus via Poincaré–Bendixson [duplicate]
Using the Poincaré–Bendixson theorem, how I can prove that the following system of polynomial ODEs
$$\begin{aligned}
\dot{x_1}=x_1-x_2-x_1^3\\
\dot{x_2}=x_1+x_2-x_2^3
\end{aligned}$$
has one limit ...
- 33
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vote
1
answer
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Converting a Cartesian ODE to Polar
I am having trouble trying to convert a first order ODE into polar coordinates.
My ODE system is as follows:
$$\frac{dx}{dt} =y$$
$$\frac{dy}{dt} = -p(x^2+y^2 -1)y - y$$
where $p$ is a parameter ...
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Terminology for closed limit set which is not a cycle?
Is there an appropriate name for a set $C$ of points in a dynamical system with all the properties:
$C$ is a closed set with the topology of $S^1$
Trajectories within $C$ remain in $C$
$C$ is the ...
- 16k
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How to show the existence of limit cycles in $\mathbb{R}^n$?
If we consider the following system:
$$
\frac{ \mathrm{d} \vec{x}}{ \mathrm{d} t}=\vec{f}(\vec{x})\qquad \text{ with } \vec{x}\in\mathbb{R}^n
$$
and assume we know a fixed point $\vec{x^*}$, we ...
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A vector field without a stationary point or a limit cycle
Background: I am a newbie to nonlinear dynamical systems. But I have taken a graduate course in linear dynamical system.
Question: Is it possible to construct an autonomous $C^1$ vector field $f:U\...
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Find all limit cycles of a system of differential equations
I am currently trying to study for an exam and ran into a problem regarding limit cycles.
The question is to find all limit cycles of the following system of the differential equations:
$$\dot{x}=-y-...
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Bifurcation analysis, limit cycle collapses on two symmetric fixed points
Coming back on the system I already mentioned in another post, this time I am working on some bifurcation analysis of a 2D System.
The system is defined by the following equations. I am assuming $\...
- 257
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Hopf Bifurcation Theorem
I have a 2D dynamical system of the form
\begin{cases}
\dot{x}=f(x,y,K) \\[1ex]
\dot{y}=g(x,y,K)
\end{cases}
where $K$ is a free parameter (later I can write the system here). I've found two Hopf ...
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Is there a proof for this limit cycle equilibrium
Consider a system of differential equations where both are both continuous partial derivatives. Let's call them $F$ and $G$. Is there a proof suggesting that if there exists a solution that is a limit ...
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Differential Equations Proof for Equilibrium
Q1
I haven been told that the system of Differential Equations are continuous partial derivatives, and also there exists a solution l(t) that is a limit cycle of the system. I have to prove why this ...
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Non-existence of limit cycle of a polynomial system
I have been assigned a project in which I need to study the following system:
$$\begin{cases}\dot{x} = x(ax^n + by^n + c)\\\dot{y} = y(dx^n + ey^n + f)\end{cases}$$
where $(a,b,c,d,e,f) \in \mathbb{R}^...
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If $L$ is a strict Lyapunov function for a planar system, then there are no limit cycles
I'm trying to follow a proof of the above lemma. My (attempted) clean up is as follows, and I will outline where I think we may have an issue.
Let $\phi_t(X)$ denote the through $X$ at time $t$.
Let $...
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Show that a system of differential equations has a periodic solution
We define a system of differential equations by
\begin{align}
\frac{dx}{dt} &= x + y − x(x^2 + 3y^2)
\\
\frac{dy}{dt} &= −x + y − 2y^3.
\end{align}
We want to show that there exists a periodic ...
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