On the Appearence of Resonant Dynamics in the Pulse Control System for the Energy Supply of a Heating Plant for Growing Sapphire Crystals

Abstract

In this paper we study the nonlinear phenomena that can be observed in an energy supply pulse modulated control system of a heating unit. The behavior of such а system is described by nonautonomous differential equations with discontinuous right-hand sides. We reduce the investigation of this system to the studying of a two-dimensional non-smooth map. We demonstrate how a quasiperiodic dynamics can arise from a stable periodic motion through a Neimark-Sacker bifurcation. The paper also discusses the specific features of the transition from phase-locked dynamics to quasiperiodicity. The regions of phase-locked dynamics dynamics in the parameter space form the so-called Arnold tongues. For piecewise smooth systems, Arnold tongues have a specific sausage-like structure. Within each resonance tongue there is an attracting closed invariant curve. This closed curve includes two cycles, a saddle and a stable, and is formed by the saddle-node connection composed of the unstable manifolds of the saddle cycle. We show that transition from a quasiperiodic to the resonance dynamics may occur in a homoclinic bifurcation. Our numerical analysis shows that firstly a pair periodic orbits (stable) appears in a saddle-node bifurcation. Near the saddle-node bifurcation point we observe the coexistence of the stable cycle and the stable closed invariant curve with a quasiperiodic dynamics. The unstable manifolds of the saddle cycle separate the basins of attraction of the coexisting motions. As the parameters change the manifolds of the saddle cycle become tangent to each other, and this leads to the formation of a nontransversal homoclinic orbit. With the further change of parameters the stable and unstable manifolds of the saddle cycle intersect transversally to form the homoclinic structure. Finally, after the second homoclinic bifurcation a stable resonant closed curve appears, which is formed by the unstable manifolds of the saddle cycle.

Acknowledgements: The work of Gol’tsov Yu.A. and Kizhuk A.S. were supported within the framework of the Program «Priority 2030» on the base of the Belgorod State Technological University named after V.G. Shukhov. The work was realized using equipment of High Technology Center at BSTU named after V.G. Shukhov.

Abdirasulov A.Z. was supported by the grant 14-22 of the Osh State University.

Yanochkina O.O. and Kolomiets E.A. were supported by the Ministry of Education and Science of the Russian Federation within the scope of the Grant «Implementation of the Strategic Academic Leadership program Priority 2030», projects No 1.71.23 П and No 1.7.21/S-2.

The research was performed under the guidance of Prof Zh. T. Zhusubaliyev, International Scientific Laboratory for Dynamics of Non-Smooth Systems, Southwest State University, Russia.