Try x'' = ax - x^3 + (1 - x'^2)x' in the vicinity of a = 1.87525 (the Rayleigh-Duffing two-well oscillator).

No, it's not possible. Try to sketch what happens, when the two limit cycles just "kiss".

Is it at all possible to have such pair of stable cycles on a plane without additional invariant manifold between them?

Interesting! As if you are imagining the collision and merging of two black holes or two tornadoes.

Yes, but the proposal is with both attracting cycles turning clockwise, which cannot happen without a zone/area with opposite winds, unless appearing a stationary submanifold(s) or discontinuities.

Check out the strange merging phenomenon of two whirlpools. There is a video too.

Scientists Observe Strange Double Whirlpool Effect in Ocean for First Time

https://gizmodo.com/scientists-observe-strange-double-whirlpool-effect-in-o-1821632281

https://i.kinja-img.com/gawker-media/image/upload/s--gI4XX2UD--/c_scale,fl_progressive,q_80,w_800/qpizwxgnvcsv3m35krb7.mp4

I don't want to regard any discontinuities in the field. And yes, there is a separating line between cycles. Maybe this picture can help to better understand the situation.

Yes, we have: Bautin codimension 2 bifurcation.

Try x'' = ax - x^3 + (1 - x'^2)x' in the vicinity of a = 1.87525 (the Rayleigh-Duffing two-well oscillator).

Thanks Prof. Julien Clinton Sprott for sharing the interesting oscillator. The solutions of x'' = (1.875 + 0.001*sin(t))*x - x^3 + (1 - (x')^2)*x', are plotted using Wolfram|Alpha.

I do not know for sure, but perhaps such a metamorphosis may occur as a result of two bifurcations.

The first is the saddle separatrix bifurcation, as a result of which three cycles appear. Two unstable inside the separatrix loop and one stable outside the separatrix loop.

When the parameter changes, the amplitude of the two unstable cycles decreases, and the amplitude of the two stable cycles increases. Then the twin limit cycle bifurcation occurs, as a result of which the unstable and stable cycles merge, destroying each other.

There remains one stable cycle that covers both the saddle and both foci (see Figures).

Excuse me for the quality of the Figures, but I'm on the road, and the question seemed too interesting to me.

That's an interesting idea about the merging of stable and unstable limit cycles. Assuming such an unstable limit cycle is a repellor that becomes an attractor in reverse time, I looked at the reversed-time system x'' = -ax + x^3 + (1 - x'^2)x' in the vicinity of a = 1.87525, and the only attractors I see are the off-axis equilibrium points. Nor do I see any conditions under which there are three stable limit cycles, and so I think that picture may not apply to that particular system, although there may be other examples to which it does.

It is certainly interesting to observe how the limit cycle changes over time.

x'' = (1.875*sin((pi/100)*t))*x - x^3 + [1 - (x')^2]*x'

one dof systems with Duffing nonlinearity and van der Pol nonlinearity may have this kind of bifurcation.

Hi, the 1:2 resonance of NS bifurcation may show this situation.

Dear Mr. Haeri,

My guess is that any statistical differential equation system representing system dynamics has power of five or more (three for one modular system) can have such limit cycles which can have systems classification and systems integration property which has a Hopf bifurcation solution as we have derived experimentally (MHM Functor Algebra Calculus on [email protected] | The Econometric Society

https://www.econometricsociety.org/users/earl-fitzpatrick-york-mallick-sir-asutosh-mukherjee-university-institute-chair-professor, in Lie and Riemann Algebra). Please refer to our paper on Market Finance for e.g. on www.researchgate.net/profile/Soumitra_Mallick.

S.K.Mallick (corresponding author)

for S.K.Mallick, S.Raychaudhury, S.Mallick

of RHMHM School,

USA, Japan, India