Quasi-periodicity in a temperature field control system of a heating unit

Author(s):  Zh.T. Zhusubaliyev, Southwest State University, Kursk, Russia, zhanybai@gmail.com

V.G. Rubanov, Dr., Prof., Belgorod State Technological University named after V.G. Shoukhov, Belgorod, Russia, vgrubanov@gmail.com

Yu.A. Gol’tsov, Belgorod State Technological University named after V.G. Shoukhov, Belgorod, Russia, umin@mail.ru

O.O.Yanochkina, Southwest State University, Kursk, Russia, yanoolga@gmail.com,

S.A. Polyakov, Southwest State University, Kursk, Russia, sergpol@yandex.ru

Issue:  Volume 44, №23

Rubric:  Computer simulation history

Annotation:  In this paper we study the nonlinear phenomena that can be observed in a temperature field 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 piecewise-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. 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. As the parameters of the system are changed, the stable and saddle cycles collide and disappear in a bordercollision fold bifurcation. In this way the resonance tongues in piecewise-smooth systems are bounded by border-collision fold bifurcation curves, rather than by the saddle-node bifurcation lines known from smooth systems

Keywords:  temperature field control system of a heating unit, quasi-periodicity, piecewise-smooth map, Neimark-Sacker bifurcation, border-collision bifurcation, two-dimensional torus, closed invariant curve

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