If you are looking for perodic solution without simulation you can find it by "periodic shooting" or by a finite difference method. I prefer the latter, where you approximate the derivatives by a finite difference and then you discretize time over the period T in N time steps. With your M first order ODEs for M state variables you can now set up M*(N-1) nonlinear algebraic equations with N*M unknowns namely the M state variables times the N time steps. The final M equations to close the problem you get from possing the requirement of periodicity x(0)=x(T). The method also works if you do not know the period T as in your example. Look in Nayfeh's book on Nonlinear Dynamics.

Sajad,

you can calculate the ellipse that approach best the limit cycle in the case of VdP at least by the method of quasilinearization. Assume x = A sin(wt) + B cos(wt) and the corresponding for y(t), replace in the ODE, and then integrate all terms over a period. Then do the same but multiplying previously by sin(wt) and also by cos(wt). You will get algebraic equations for the amplitude and w, as a functuion of the ODE parameters (a in your example).

For small a the approach is quite good. For large a the cycle departs from the ellipse, so you may try to add higher harmonics, adn take higher moments of the ODE.

Alejandro

See Zoladek paper from 1998:

Algebraic invariant curves for the Liénard equation, pp 1681-1701

H Żoładek - Transactions of the American Mathematical Society, 1998 - ams.org

See also

Kenzi Odani, The limit cycle of the van der Pol equation is not algebraic, J. Differential Equations 115 (1995), no. 1, 146–152. MR 1308609 (95i:34051), http://dx.doi.org/10.1006/jdeq.1995.1008

To solve the differential equation:

dy/dx = a(1-x^2) - x/y

one may use various numeric (or perhaps analytic) solution methods.

The solution found by a numeric method can be fitted by different sets of orthogonal functions such as the Fourier series.

Hale and Kocak Bifurcations and Chaos book has an analytic proof of the shape of the limit cycle in Van der Pol. It's not pretty...

the shape is mainly done by slow manifold.

Ask professor Jean-Marc Ginoux (he is affiliated to researchgate), he developped a new method in order to compute slow manifold in any dimension and any case.

This may not be exactly what you are looking for: but might help indirectly to find the info you wish:

B. V. Dasarathy, “On the Forced Response of a Class of Van Der Pol - Duffing Oscillator,” Proceedings of the IEEE, Vol. 58, No. 4, pp. 602-603, April 1970. (I)

If you find an inverse integrating factor V(x,y) then the limit cycle must be contained into its zero set V^{-1}(0)

You can perform successive estimation based on harmonic balance (i.e. a Fourier series like expansion) and get a very close resemblance to the real trajectory without simulation. One needs to know the period however.

This used to be the technique in the XIX century for planetary orbit computations prior to digital simulation.

Check out the book by Minorsky, Nonlinear Oscillations, for example.

I hope this helps,

Best wishes,

Philippe Mullhaupt

Dear Sajad Jafari,

As far as I know there is not a fully algebraic approach for this problem.

However, it is a standard question how to do this numerically by using Floquet multipliers. This method has been described at length in, e.g.,

- Apri et al, Efficient estimation of the Robustness region of Biological models with oscillatory behaviour, PLoS One 5(4): e9865 (2010), DOI: 10.1371/journal.pone.0009865

- See also the standard book by Nayfeh et al (2004): Applied nonlinear dynamics, Wiley-VCH

Success, Jaap Molenaar

If you are looking for perodic solution without simulation you can find it by "periodic shooting" or by a finite difference method. I prefer the latter, where you approximate the derivatives by a finite difference and then you discretize time over the period T in N time steps. With your M first order ODEs for M state variables you can now set up M*(N-1) nonlinear algebraic equations with N*M unknowns namely the M state variables times the N time steps. The final M equations to close the problem you get from possing the requirement of periodicity x(0)=x(T). The method also works if you do not know the period T as in your example. Look in Nayfeh's book on Nonlinear Dynamics.

Kiselev Yuri, Moscow Lomonosov University

To find limit cycle you can use construct boundary value problem for ODE

which is solved numerically (continuation approach). Example of solving such a problem

(limit cycles for Equailer system)

may be found in the hand book

Kiselev Yu.N., Avvakumov S.N., Orlov M.V. Optimal control. Linear theory and applications. Moscow. 2007, (In Russian).

See

http://www.twirpx.com/file/137468/

The figure seems to correspond to large values of a. After rescaling time the equation goes to the vdP oscillator with small inertia. There are several constructions of the limit cycle using matched asymptotic expansions for small inertia in J. Grasman's book Asymptotic methods for relaxation oscillations and applications. Springer 1987.

In general, it is usually impossible to find an analytic expression for the limit cycle.

In particular, an explicit formula for the limit cycle of the Van der Pol system is apparently not known but in the paper https://www.researchgate.net/publication/248819427_The_Limit_Cycle_of_the_van_der_Pol_Equation_Is_Not_Algebraic it is shown that it is *not* algebraic.

Article The Limit Cycle of the van der Pol Equation Is Not Algebraic

This is a slow fast system. What you want is in O'Maly book "perturbation method..."

He find the limit cycle analitically and calculate it's period.

Some of the above answers are misleading, because the correct answer is

NO.

It is possible to construct artificial exceptions when it can be determined. Speaking about approximations I mention that the period of the oscillatory solution can symbolically (not nemuerically) be approximated, see e.g. Farkas: Periodic Motions, Springer.

There was an interesting text that I read some 40 years ago - Nonlinear oscillations by Nicolai Minorsky. Some of the asymptotic methods provided useful characterizations. I found a relatively simple summary of his work in:

http://www.tuks.nl/pdf/Reference_Material/Minorsky/Minorsky%20-%20Theoretical%20Aspects%20of%20Nonlinear%20Oscillations%20-%201960.pdf

You might find this interesting.

http://www.uk.ac.ir/fs/global/modulesContent/html/editor/1375/8.pdf

I was going to suggest that you use perturbation methods to get an approximate analytical solution, then get a numerical initial condition on the limit cycle and use that as an IC to get the limit cycle, but if a=2, there is no really "small" parameter..

It also seems you could solve by series methods, and use an iterative scheme to build up successive terms using a numerically obtained IC on the limit cycle, refining your estimate of IC and series terms based on how close you come to closing the curve.

Perturbation methods find an "exact" solution in a certain sense, in that successive approximating equations may converge to an exact solution. Here is a nice example on the VDP

http://www.maths.qmul.ac.uk/~tp/pujeh_thesis.pdf

The best tool for this purpose is, in my view, the Poincare normal form. Look it up in one of the Arnold's book, or, in a more accessible form with a number of applications, including PDE, in Kuznetsov's book "Elements of applied bifurcation theory".

In general, no, there is no way to find an exact representation of limit cycles. There are a number of techniques to approximate the shape (several noted above). What do you want the representation for? If you're just going to do a visual examination I think that you'll find that simulation is the easiest route (unless the system at hand has some pathology). If you're looking to have some analytical expression for use in some subsequent computations the appropriate choice of method is going to be dependent upon the particulars of your system and the precision and accuracy requirements. So-called small parameter problems are good candidates for perturbation methods, if f is analytic, you might want to look at Carleman linearization (if the system satisfies the Poincare-Sternberg theorem, you can apply Poincare linearization which will give you what you appear to be seeking), discontinuous methods can be used for piecewise analyses of larger parameter and discontinuous systems as can asymptotic methods. Note that larger parameter systems often display quasi-discontinuity and power series solutions are unavailable.) If it's the van der Pol equation that you are interested, most of these (and other) methods have been applied to it in the literature.

Theoretically speaking, we can locate limit cycles for systems that undergoes Hopf Bifurcation: Reduce the system on the center manifold, then find the simplest normal forms for the system and, finally, transform the obtained normal forms using polar coordinates. Because the Hopf bifurcation allow us to determine a critical value, of any parameter of our system, for which a limit cycle must exist around it! If you are so so lucky, the simplest normal form for your system could allow to construct analytically this limit cycle. Otherwise, you can only locate this limit cycle. The set of dynamical systems for which you can construct the a periodic solutions are so so rare!

- prove that the system is almost linear.

- find the eigenvalues of the Jacobian matrix. Or,

- You can apply Liapunov variable - Gradient Method to get an explicit formula relates the two dependent variables, x and y.

instead of all these steps, u may solve the system numerically.

In the study of the nonlinear flutter instability of thin plates under nonconservative loads, it was observed that, under certain loads, the limit cycle has the form of a Lamé curve (super-ellipse of real order). See: John T. Katsikadelis And Nick G. Babouskos, Nonlinear Flutter Instability Of Thin Damped Plates: A Solution By The Analog Equation Method, Journal Of Mechanics Of Materials And Structures, Vol. 4, No. 7-8, 2009. It would be interesting to investigate this issue theoretically.

Thank you Manuel, I will investigate the materials you introduced.

Sajad, please tell me later if you reach the answer, if not I will write the concrete example and I will put here in researchgate...

Sajad, I have written the concrete answer to your question such I told you some months agos, please check: https://www.researchgate.net/publication/264861771_Solucion_explicita_al_ciclo_de_la_ecuacion_de_van_der_Pol_de_tercer_grado_%28provisional%29?ev=prf_pub

Working Paper Solucion explicita al ciclo de la ecuacion de van der Pol de...

Whilst as a phd student I worked on a method do such calculations to derive a functional form in the phase plane i.e. f(y',y) = 0. which when plotted gives closed orbits with very little departure from the limit cycle orbit. Furthermore, I used the van der pol as a benchmark case for the method, along with my own nonlinear ode that exhibits limit cycle behavior. If you contact me I can provide further information (my thesis which outlines the approach). Cheers

Answer is available by Poincare Section analysis based on events.  attached file is your answer.

As it is indicated by Prof. Artur Sergyeyev, the limit cycle of the Van der pol equation is not algebraic hence does not admit an explicit and analytic representation.

In fact "Differential Galois theory" is a tool to show that almost all polynomial vector field do not admit an explicit representation for their solution.

Limit cycles are very mysterious and unreachable objects. They are the main subject of the second part of the Hilbert 16th problem.

In the following project I try to find some qualitative (but not numerical) methods to approach these objects.

https://www.researchgate.net/project/Two-projects-related-to-limit-cycle-theory