Consider a general 2D system:
x' = f(x,y)
y’ = g(x,y)
Do you know any such system (preferably a simple one, ideally quadratic) which has more than one limit cycle? I would prefer it if there was at most one unstable equilibrium.
The general problem of finding the number of limit cycles for a specific dynamical system is related to the Hilbert’s 16th problem. Among the dynamical systems with several limit cycles you can find Lienard systems, being the van der Pol system a particular case. Some references and examples can be found in:
M. A. F. Sanjuan, Lienard Systems, Limit Cycles, and Melnikov Theory”, Physical Review E57(1), 340-344 (1998)
There is a non-countable number of systems with that property of having more than one limit cycle. Van Der Pol is not quadratic.
Edited: I mentioned Van Der Pol since the author referred Van Der Pol as an instance of a quadratic map. After my answer that part of the question was removed from the question by the author.
Dear Henrique,
I didn't mention Van Der Pol as a quadratic map. I mentioned it as a system which has only one limit cycle. I removed it because it caused misunderstanding.
Can you please mention one of those non-countable number of systems?
Thank you
Ok, ok. Any system of the type:
r'=A f(r)
θ'=Const.
Can have as many as you please limit cycles depending on the zeros of f. It can have even an infinite number of limit cycles. The number of such systems in non-countable. A trivial change of variables gets what you want.
Best regards
H
Or
x'=Cy - K x f(r)/r
y'=-Cx- K y f(r)/r
r=Sqrt(x²+y²)
With f(r) with the desired number of roots. The stability is also easy.
x''+x+ epsilon*Sin[x']=0
x''+x+epsilon*x'*Cos[x]=0
x''+x+epsilon*x'^3*Cos[x]=0
See discussion in book by Kahn & Zarmi
The general problem of finding the number of limit cycles for a specific dynamical system is related to the Hilbert’s 16th problem. Among the dynamical systems with several limit cycles you can find Lienard systems, being the van der Pol system a particular case. Some references and examples can be found in:
M. A. F. Sanjuan, Lienard Systems, Limit Cycles, and Melnikov Theory”, Physical Review E57(1), 340-344 (1998)
Dear Sajad,
In case you are still interested in a simple example of a 2D flow with more than one limit cycles, you could have a look in the paper in:
http://arxiv.org/pdf/1001.3240v1.pdf
the modified van der Pol oscillator eq. (1) has from one to three limit cycles; in the latter case, two of them are stable and one unstable!
Regards,
Nikos
Dear Sajad
There is a quadratic system with 4 Limit cycles please see
http://planetmath.org/hilberts16thproblemforquadraticvectorfields
The method of construction of a quadratic system with 4 limit cycles is based on the following:
They pick a quadratic sytem with 2 hyperbolic limit cycles with disjoint interior. Existence of these 2 limit cycles is a consequence of Poincare-Bendixson theorem. One of thses 2 limit cycles suround a weak focus of order 2. So after perturbation we obtain two additional (smal) limit cycles. So we have 4 limit cycles with (3,1) distribution
It is unknown whether there is a quadratic system with more than 4 limit cycle. It is also unknown that whether there is a (2,2) distribution of limit cycles for quadratic systems.
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