Avram Iancu University

Avram Iancu, Romania

Avram Iancu University

Avram Iancu, Romania
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Danca M.-F.,Avram Iancu University | Danca M.-F.,Romanian Institute of Science and Technology
Nonlinear Dynamics | Year: 2017

In this paper, we present a scheme for uncovering hidden chaotic attractors in nonlinear autonomous systems of fractional order. The stability of equilibria of fractional-order systems is analyzed. The underlying initial value problem is numerically integrated with the predictor-corrector Adams-Bashforth-Moulton algorithm for fractional-order differential equations. Three examples of fractional-order systems are considered: a generalized Lorenz system, the Rabinovich-Fabrikant system and a non-smooth Chua system. © 2017 Springer Science+Business Media Dordrecht


Danca M.-F.,Avram Iancu University | Danca M.-F.,Coneural Center for Cognitive and Neural Studies
Nonlinear Analysis: Real World Applications | Year: 2010

The study of uniqueness of solutions of discontinuous dynamical systems has an important implication: multiple solutions to the initial value problem could not be found in real dynamical systems; also the (attracting or repulsive) sliding mode is inherently linked to the uniqueness of solutions. In this paper a strengthened Lipschitz-like condition for differential inclusions and a geometrical approach for the uniqueness of solutions for a class of Filippov dynamical systems are introduced as tools for uniqueness. Several theoretical and practical examples are discussed. © 2009 Elsevier Ltd. All rights reserved.


Danca M.-F.,Avram Iancu University | Danca M.-F.,Institute of Science and Technology
International Journal of Bifurcation and Chaos | Year: 2011

In this letter we synthesize numerically the Lü attractor starting from the generalized Lorenz and Chen systems, by switching the control parameter inside a chosen finite set of values on every successive adjacent finite time intervals. A numerical method with fixed step size for ODEs is used to integrate the underlying initial value problem. As numerically and computationally proved in this work, the utilized attractors synthesis algorithm introduced by the present author before, allows to synthesize the Lü attractor starting from any finite set of parameter values. © 2011 World Scientific Publishing Company.


Danca M.-F.,Avram Iancu University | Danca M.-F.,Romanian Institute of Science and Technology
Dynamical Systems | Year: 2010

In this article we prove numerically, via computer graphic simulations and specific examples, that switching the control parameter of a dynamical system belonging to a class of dissipative continuous dynamical systems, one can obtain a stable attractor. In this purpose, while a fixed step-size numerical method approximates the solution of the mathematical model, the parameter control is switched every few integration steps, the switching scheme being time periodic. The switch occurs within a considered set of admissible parameter values. Moreover, we show via numerical experiments that the obtained synthesized attractor belongs to the class of all admissible attractors for the considered system and matches to the averaged attractor obtained with the control parameter replaced with the averaged switched parameter values. This switched strategy may force the system to evolve along on a stable attractor whatever the parameter values and introduces a convex structure inside of the attractor set via a bijection between the set of parameter control values and the attractors set. The algorithm besides its utility in systems stabilization, when some desired parameter control cannot be directly accessed, may serve as a model for the dynamics encountered in reality or in experiments, e.g. three species food chain models, electronic circuits, etc. This method, compared, for example, to the OGY algorithm where only small perturbations of parameter control can be issued, allows relatively large parameter perturbations. Also, it does not allow to stabilize an unstable orbit but, using an appropriate parameter switching algorithm, it allows to reach an already existing attractor. The present work extends the results we obtained previously and is applied to Lorenz, Rössler and Chen systems. © 2010 Taylor & Francis.


Danca M.-F.,Avram Iancu University | Danca M.-F.,Romanian Institute of Science and Technology
Nonlinear Dynamics | Year: 2010

In this paper, the chaos persistence in a class of discontinuous dynamical systems of fractional-order is analyzed. To that end, the initial value problem is first transformed, by using the Filippov regularization (Filippov in Differential Equations with Discontinuous Right-Hand Sides, 1988), into a set-valued problem of fractional-order, then by Cellina's approximate selection theorem (Aubin and Cellina in Differential Inclusions Set-valued Maps and Viability Theory, 1984; Aubin and Frankowska in Set-valued Analysis, 1990). The problem is approximated into a single-valued fractional-order problem, which is numerically solved by using a numerical scheme proposed by Diethelm et al. (Nonlinear Dyn. 29:3-22, 2002). Two typical examples of systems belonging to this class are analyzed and simulated. © 2009 Springer Science+Business Media B.V.


Danca M.-F.,Avram Iancu University | Danca M.-F.,Romanian Institute of Science and Technology
Computers and Mathematics with Applications | Year: 2012

In this paper, we prove that an existing algorithm with periodical impulses for chaos suppression in continuous systems can be further extended and successfully applied to a class of piecewise continuous systems. For this purpose, the underlying initial value problem is transformed into a continuous problem via Filippov regularization and Cellina's approximation theorem. Next, some results on existence and continuation for ODEs with impulses are applied. As an example, the piece-wise continuous Chen system is considered. © 2011 Elsevier Ltd. All rights reserved.


Danca M.-F.,Avram Iancu University | Danca M.-F.,Romanian Institute of Science and Technology
International Journal of Bifurcation and Chaos | Year: 2011

In this paper, we are concerned with the possible approach to the existence of solutions for a class of discontinuous dynamical systems of fractional order. To this purpose, the underlying initial value problem is transformed into a fractional set-valued problem. Next, Cellina's Theorem is applied leading to a single-valued continuous initial value problem of fractional order. The existence of solutions is assured by a Péano-like theorem for ordinary differential equations of fractional order. © 2011 World Scientific Publishing Company.


Danca M.-F.,Avram Iancu University | Danca M.-F.,Romanian Institute of Science and Technology
Nonlinear Dynamics | Year: 2016

In Danca et al. (Int J Bifurc Chaos 26(02):1650038, 2016), it is shown that the Rabinovich–Fabrikant (RF) system admits self-excited and hidden chaotic attractors. In this paper, we further show that the RF system also admits a pair of symmetric transient hidden chaotic attractors. We reveal more extremely rich dynamics of this system, such as a new kind of “virtual saddles.” © 2016 Springer Science+Business Media Dordrecht


Danca M.-F.,Avram Iancu University | Danca M.-F.,Romanian Institute of Science and Technology
Nonlinear Dynamics | Year: 2014

This paper proves analytically that synchronization of a class of piecewise continuous fractional-order systems can be achieved. Since there are no dedicated numerical methods to integrate differential equations with discontinuous right-hand sides for fractional-order models, Filippov’s regularization (Filippov, Differential Equations with Discontinuous Right-Hand Sides, 1988) is applied, and Cellina’s Theorem (Aubin and Cellina, Differential Inclusions Set-valued Maps and Viability Theory, 1984; Aubin and Frankowska, Set-valued Analysis, 1990) is used. It is proved that the corresponding initial value problem can be converted to a continuous problem of fractional-order systems, to which numerical methods can be applied. In this way, the synchronization problem is transformed into a standard problem for continuous fractional-order systems. Three examples are presented: the Sprott’s system, Chen’s system, and Shimizu–Morioka’s system. © 2014, Springer Science+Business Media Dordrecht.


Danca M.-F.,Avram Iancu University | Danca M.-F.,Romanian Institute of Science and Technology
Nonlinear Dynamics | Year: 2011

In this paper we investigate the possibility to formulate an implicit multistep numerical method for fractional differential equations, as a discrete dynamical system to model a class of discontinuous dynamical systems of fractional order. For this purpose, the problem is continuously transformed into a set-valued problem, to which the approximate selection theorem for a class of differential inclusions applies. Next, following the way presented in the book of Stewart and Humphries (Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996) for the case of continuous differential equations, we prove that a variant of Adams-Bashforth-Moulton method for fractional differential equations can be considered as defining a discrete dynamical system, approximating the underlying discontinuous fractional system. For this purpose, the existence and uniqueness of solutions are investigated. One example is presented. © 2010 Springer Science+Business Media B.V.

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