System approach of solving direct and reverse tasks in systems with chaos
Keywords:deterministic chaos, YU.-SH. Chen nonlinear system, bifurcation, reconstruction of mathematical model
AbstractIn this paper, the systematic approach to the effective application of mathematical and computer modeling of dynamic systems is proposed for solving the problems of deterministic chaos research in complex nonlinear systems and related inverse problems. The scientific and technical task of enhancing mathematical modeling by improving existing methodologies of investigation of the deterministic chaos and by developing new mathematical models, based on the specialization of existing ones, is solved. To solve the problem, we suggested investigation schemes of direct (research modes of behavior depending on the bifurcation parameters) and inverse (reconstruction of mathematical models) tasks of the deterministic chaos in complex non-linear systems. Experimental studies are presented for scalar implementations of YU.-SH. Chen and Roessler nonlinear systems. For the last one, the equivalent model was constructed.
Krasnopol'skaya T.S. Regular and chaotic surface waves in a liquid in a cylindrical tank / T.S Krasnopol'skaya, A.Yu. Shvets // Soviet Applied Mechanics, 1990. – Vol. 26. – № 8. – P. 787–794.
Shvets A.Yu. Chaotic Oscillations of Nonideal Plane Pendulum Systems / A.Yu. Shvets, A.M. Makaseyev // Chaotic Modeling and Simulation (CMSIM) Journal, 2012. – N 1. – P. 195–204.
Zinchenko A.Ju. Issledovanie reguljarnoj i haoticheskoj dinamiki odnoj finansovoj sistemy / A.Ju. Zinchenko // Izvestija vysshih uchebnyh zavedenij. Prikladnaja nelinejnaja dinamika. – 2013. – T. 21, № 2. – S. 173–187.
Crutchfield J.P. Equations of motion from a data series / J.P. Crutchfield, B.S. McNamara // Complex Systems. – 1987. – N 1. – P. 417–452.
Ma J.H. Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system. I / J. H. Ma, Y.S. Chen // Applied Mathematics and Mechanics. – 2001. – Vol. 22, N 11. – P. 1240–1251.
Dormand J.R. A family of embedded Runge-Kutta formulae / J.R. Dormand, P.J. Prince // J. Comp. Appl. Math. – 1980. – Vol. 6. – P. 19–26.
Danylov V.Ja. Synerhetychni metody analizu: navch. posib. / V.Ja. Danylov, A.Ju. Zinchenko. – K.: NTUU "KPI" VPI VPK "Politekhnika", 2011. – 340 s.