# 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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### 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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### 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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### 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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### 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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