Complex dynamics of a modified four order Wien-bridge oscillator model and FPGA implementation
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Complex dynamics of a modified four order Wien-bridge oscillator model and FPGA implementation Herman Landry Ndassi1,2 , Achille Ecladore Tchahou Tchendjeu2,3 , Marceline Motchongom Tingue4,a , Edwige Raissa Mache Kengne1, Robert Tchitnga1,2 , Martin Tchoffo1 1 Unité de recherche de Matière Condensée d’Electronique et de Traitement du Signal (URMACETS), Faculty
of Sciences, University of Dschang, P.O.Box 67, Dschang, Cameroon
2 Research Group on Experimental and Applied Physics for Sustainable Development (EAPhySuD),
P.O.Box 412, Dschang, Cameroon
3 Department of Computer Engineering, National Higher Polytechnic Institute, The University of Bamenda,
P.O.Box 39, Bambili, Cameroon
4 Present Address: Higher Technical Teachers Training College, The University of Bamenda, P.O. Box 39,
Bambili, Cameroon Received: 11 April 2020 / Accepted: 4 September 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper presents a novel fourth-order autonomous flux controlled memristive Wien-bridge system. Standard nonlinear diagnostic tools such as bifurcation diagram, graphs of largest Lyapunov exponent, Lyapunov stability diagram, phase space trajectory and isospike diagram are used to characterize dynamics of the system. Results show that the system presents hidden attractors with infinite equilibrium points known as line equilibrium for a suitable set of its parameters. The system also exhibits striking phenomenon of extreme multistability. Through isospike and Lyapunov stability diagram, spiral bifurcation leading to a center hub point is observed in a Wien-bridge circuit for the first time and the Field Programmable Gate Array -based implementation is performed to confirm its feasibility.
1 Introduction Since the discovery of chaos, systems with chaotic dynamics have been modeled and implemented using nonlinear circuits. Their applications have been developed in many engineering domains such as robotics [1], cryptosystems [2] and neuroscience [3, 4], just to name a few. Known as a fourth basic circuit element, the memristor was postulated in 1971 by Chua [5] and later implemented (as a nanoelectronic TiO2 device) in Hewlett–Packard (HP) laboratory by Stanley Williams and his team [6]. Interestingly, the memristor model have two types of implementation circuits with the off-the-shelf discrete components, namely the operational amplifier and analog multipliers–based equivalent circuits [7–10] and memristive diode bridges cascaded with L, RC or LC components [11–13]. Just a few years ago, several memristor–based systems have been used in different domains such as memristive adaptive [14], memristive model of amoeba learning [15], and resistance switch-
a e-mail: [email protected] (corresponding author)
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ing memories [16], memristor cellular [17], or programmable analog integrated circuits [18]. Rich dynamical behaviors leading to striking applications with memristive based syste
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