Generating Chaos in 3d Systems of Quadratic Differential Equations With 1d Exponential Maps

Abstract

New existence conditions of homoclinic orbits for some systems of ordinary quadratic differential equations with singular linear part are found. A realization of these conditions guarantees the existence of chaotic attractors at 3D autonomous quadratic systems. In addition, a chaotic behavior of solutions of these systems is determined by the 1D discrete map

Role of logistic and Ricker’s maps in appearance of chaos in autonomous quadratic dynamical systems

“…The state of chaos of the map f (v) on the interval [0, 1] can be proved by the methods offered in [22]. Thus, the conclusions of all items (d1)-(d5) allow to complete the proof of Theorem 5 and its Corollary.…”

“…Then from here the discrete process (10) may be derived. The state of chaos of the map (17) on the interval [0, ∞) was proved in [22].…”

“…The state of chaos of the map f (ρ) = ρ · exp(G + Fρ − Eρ 2 ) on the interval [0, ∞) can be proved by the methods offered in [22].…”

“…Here basic results are contained, for example, in [20][21][22][23][24][25][26][27][28]. The main idea of these papers is that properties of the being created discrete maps, which describe a behavior of continuous dynamical systems, are based on the well-known properties of the Ricker map f (x) = x exp(r − x) or the logistic map g(x) = r x(1 − x).…”

General Method of Construction of Implicit Discrete Maps Generating Chaos in 3D Quadratic Systems of Differential Equations

“…However, since they can be widely applied, more chaos systems need be found or created actively to meet the needs of engineering applications (see, for example, [Belozyorov, 2011b;Belozyorov, 2012;Belozyorov & Chernyshenko, 2013;Feng & Tse, 2007;Khan & Kumar, 2013;Kocan, 2012;Luo & Guo, 2010;Robinson, 2004;Shen & Jia, 2011;Zhang et al, 2009], and many references cited therein).…”