Chaos transition of the generalized fractional duffing oscillator with a generalized time delayed position feedback
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ORIGINAL PAPER
Chaos transition of the generalized fractional duffing oscillator with a generalized time delayed position feedback Mohamed El-Borhamy
Received: 12 June 2019 / Accepted: 22 July 2020 © Springer Nature B.V. 2020
Abstract This article is concerned with the study of chaos transition of the duffing oscillator in the presence of fractional-order derivative and third-order polynomial delayed position feedback. The main aim of the work is focused on proving that the existence of time delay parameter has the ability to change the dynamic state of the fractional duffing oscillator from regular to chaotic in the absence of damping and external excitation. The concluded theoretical results are verified numerically using self-implemented MATLAB codes for some special cases of the fractional duffing oscillator with a delayed feedback. Keywords Fractional differential equations · Functional differential equations · Bifurcations · Periodic solutions · Chaotic dynamics Mathematics Subject Classification 34A08 · 34K05 · 34C23 · 34C25 · 37D45
ticular, control and vibrational systems, which have great interests in mechanical and electrical engineering designs [20,23,29,36,39]. In engineering systems, it has been noticed that the insertion of fractional derivative and history effect might be a directed starting step to simulate the real responses [12,13,50]. Thus, the existence of time delay and fractional derivative have become common approaches to take care of previous occurring dynamics and infinite dimensionality of systems, respectively. One of most important engineering dynamical systems which is playing a main nerve in the study of nonlinear vibrational systems is the damped-driven duffing oscillator [9,21]. Indeed, its fractional type is considered now the main describing model for many physical, mechanical, and even biological engineering problems to study the characteristics of their vibrational motions. In general, the classical duffing oscillator in its original case, containing two components to generate a vibrational behavior, reads:
1 Introduction
x¨ = f (x),
In recent decades, fractional delayed dynamical systems have been considered as a basic tool to describe real processes in every engineering disciplines, in par-
the first is a source of inertia, and the second is a restoring mechanism characterizing the elasticity of the system. The function f (x) might be considered as a nonlinear restoring force(per unit mass) acting alike a hard spring and typically characterized by a super-linearity at infinity [14]. However, the nonlinearity of f (x) has a big responsibility for producing complex dynamical properties exhibited by the duffing oscillator.
M. El-Borhamy (B) Department of Engineering Mathematics and Physics, Faculty of Engineering, University of Tanta, Tanta, Egypt e-mail: [email protected]; [email protected]
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M. El-Borhamy Fig. 1 Lorenz’ strange attractor and the correlation dimension Dc = 2.08
Obviously, most of the presented mathematical works of Eq.
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