The effect of symmetry breaking on reversible systems with mixed dynamics

Theme – the effect of symmetry violation on the structure of the phase space of invertible systems. Aim – to study the changes in the phase space structure of invertible systems caused by the violation of symmetry, in particular, the possibility of multistability and the types of coexisting attractors. The peculiarities in comparison with the similar regimes in the systems with fixed constant dissipation also studied. Methods – the numerical simulation of the system of coupled phase equations for four oscillators with weak coupling with different coupling functions both with symmetry and without it. The methods of phase portraits and attractors plotting, the calculation of Lyapunov exponents spectra, the search for stable and unstable cycles and the manifolds of saddle cycles are used. Results. It was shown that the violation of symmetry results in the destruction of conservative dynamics and the attractors occur. Unlike the systems with constant weak dissipation the number of coexisting attractors is small but both periodic and chaotic attractors occur. The heteroclinic structures also are revealed. Discussion – the results are rather common because of the simple nature of used system which is the model system for the wide class of systems – the chains of oscillating systems with weak coupling.