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Open AccessArticle

The Verhulst-Like Equations: Integrable OΔE and ODE with Chaotic Behavior

1
Institut für Allgemeine Mechanik, RWTH Aachen University, Templergraben 64, D-52056 Aachen, Germany
2
Dnipropetrovs’k Regional Institute of Public Administration, National Academy of Public Administration, Office of the President of Ukraine, 29, Gogol St., 49044 Dnipro, Ukraine
3
Department of Mathematical and Computer Modeling, Al-Farabi Kazakh National University, 71 al-Farabi ave., Almaty 050040, Kazakhstan
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(12), 1446; https://doi.org/10.3390/sym11121446
Received: 28 August 2019 / Revised: 15 November 2019 / Accepted: 20 November 2019 / Published: 25 November 2019
(This article belongs to the Special Issue Asymptotic Methods in the Mechanics and Nonlinear Dynamics)
In this paper, we study various variants of Verhulst-like ordinary differential equations (ODE) and ordinary difference equations (O Δ E). Usually Verhulst ODE serves as an example of a deterministic system and discrete logistic equation is a classic example of a simple system with very complicated (chaotic) behavior. In our paper we present examples of deterministic discretization and chaotic continualization. Continualization procedure is based on Padé approximants. To correctly characterize the dynamics of obtained ODE we measured such characteristic parameters of chaotic dynamical systems as the Lyapunov exponents and the Lyapunov dimensions. Discretization and continualization lead to a change in the symmetry of the mathematical model (i.e., group properties of the original ODE and O Δ E). This aspect of the problem is the aim of further research. View Full-Text
Keywords: Verhulst ODE; Verhulst OΔE; discretization; continualization; periodic motion; subharmonic; chaos; Poincaré section; Lyapunov exponent; Lyapunov dimension Verhulst ODE; Verhulst OΔE; discretization; continualization; periodic motion; subharmonic; chaos; Poincaré section; Lyapunov exponent; Lyapunov dimension
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Andrianov, I.; Starushenko, G.; Kvitka, S.; Khajiyeva, L. The Verhulst-Like Equations: Integrable OΔE and ODE with Chaotic Behavior. Symmetry 2019, 11, 1446.

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