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Marius-F. Danca

Publications and source records attributed to Marius-F. Danca.

At least 37 records · Page 2Linked to original sources
arXiv2018

Graphical exploration of the connectivity sets of alternated Julia sets; M, the set of disconnected alternated Julia sets

Using computer graphics and visualization algorithms, we extend in this work the results obtained analytically in [1], on the connectivity domains of alternated Julia sets, defined by switching the dynamics of two quadratic Julia sets. As proved in [1], the alternated Julia sets exhibit, as for polynomials of degree greater than two, the disconnectivity property in addition to the known dichotomy property (connectedness and totally disconnectedness) which characterizes the standard Julia sets. Via computer graphics, we unveil these connectivity domains which are four-dimensional fractals. The computer graphics results show here, without substituting the proof but serving as a research guide, that for the alternated Julia sets, the Mandelbrot set consists of the set of all parameter values, for which each alternated Julia set is not only connected, but also disconnected.

math.DS↗
arXiv2018

Graphical structure of attraction basins of hidden attractors: the Rabinovich-Fabrikant system

For systems with hidden attractors and unstable equilibria, the property that hidden attractors are not connected with unstable equilibria is now accepted as one of their main characteristics. To the best of our knowledge this property has not been explored using realtime interactive three-dimensions graphics. Aided by advanced computer graphic analysis, in this paper, we explore this characteristic of a particular nonlinear system with very rich and unusual dynamics, the Rabinovich-Fabrikant system. It is shown that there exists a neighborhood of one of the unstable equilibria within which the initial conditions do not lead to the considered hidden chaotic attractor, but to one of the stable equilibria or are divergent. The trajectories starting from any neighborhood of the other unstable equlibria are attracted either by the stable equilibria, or are divergent

nlin.CD↗
arXiv2018

A simplified superheating Rankine pump with possible application in irrigation

In this paper a simplified superheating Rankine pump is presented. For an easy understanding and study of its functioning principle, the pump has been experimentally implemented, with superheating obtained by an electrical resistance. The pump can be adapted to serve as a reliable and cheap irrigation pump, having reduced maintenance and operating costs, and consuming clean energy (concentrate solar energy).

physics.app-ph↗
arXiv2018

Hidden chaotic attractors in fractional-order systems

In this paper, we present a scheme for uncovering hidden chaotic attrac- tors in nonlinear autonomous systems of fractional order. The stability of equilibria of fractional-order systems is analyzed. The underlying initial value problem is nu- merically integrated with the predictor-corrector Adams-Bashforth-Moulton algo- rithm 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.

nlin.CD↗
arXiv2018

Matlab code for Lyapunov exponents of fractional order systems

In this paper the Benettin-Wolf algorithm to determine all Lyapunov exponents for a class of fractional-order systems modeled by Caputo's derivative and the corresponding Matlab code are presented. First it is proved that the considered class of fractional-order systems admits the necessary variational system necessary to find the Lyapunov exponents. The underlying numerical method to solve the extended system of fractional order, composed of the initial value problem and the variational system, is the predictor-corrector Adams-Bashforth-Moulton for fractional differential equations. The Matlab program prints and plots the Lyapunov exponents as function of time. Also, the programs to obtain Lyapunov exponents as function of the bifurcation parameter and as function of the fractional order are described. The Matlab program for Lyapunov exponents is developed from an existing Matlab program for Lyapunov exponents of integer order. To decrease the computing time, a fast Matlab program which implements the Adams-Bashforth-Moulton method, is utilized. Four representative examples are considered.

physics.comp-ph↗
arXiv2016

Hidden transient chaotic attractors of Rabinovich-Fabrikant system

In [1], 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".

nlin.CD↗
arXiv2016

Note on a parameter switching method for nonlinear ODEs

In this paper we study analytically a parameter switching (PS) algorithm applied to a class of systems of ODE, depending on a single real parameter. The algorithm allows the numerical approximation of any solution of the underlying system by simple periodical switches of the control parameter. Near a general approach of the convergence of the PS algorithm, some dissipative properties are investigated and the dynamical behavior of solutions is investigated with the Lyapunov function method. A numerical example is presented

nlin.CD↗
arXiv2016

Chaos control of Hastings-Powell model by combining chaotic motions

In this paper, we propose a Parameter Switching (PS) algorithm as a new chaos control method for the Hastings-Powell (HP) system. The PS algorithm is a convergent scheme that switches the control parameter within a set of values while the controlled system is numerically integrated. The attractor obtained with the PS algorithm matches the attractor obtained by integrating the system with the parameter replaced by the averaged value of the switched parameter values. The switching rule can be applied periodically or randomly over a set of given values. In this way, every stable cycle of the HP system can be approximated if its underlying parameter value equalizes the average value of the switching values. Moreover, the PS algorithm can be viewed as a generalization of Parrondo's game, which is applied for the first time to the HP system, by showing that losing strategy can win: "losing + losing = winning". If "loosing" is replaced with "chaos" and, "winning" with "order" (as the opposite to "chaos"), then by switching the parameter value in the HP system within two values, which generate chaotic motions, the PS algorithm can approximate a stable cycle so that symbolically one can write "chaos + chaos = regular". Also, by considering a different parameter control, new complex dynamics of the HP model are revealed.

nlin.CD↗
arXiv2016

Unusual dynamics and hidden attractors of the Rabinovich-Fabrikant system

This paper presents some unusual dynamics of the Rabinovich-Fabrikant system, such as "virtual" saddles, "tornado"-like stable cycles and hidden chaotic attractors. Due to the strong nonlinearity and high complexity, the results are obtained numerically with some insightful descriptions and discussions.

nlin.CD↗
arXiv2016

Parameter Switching Synchronization

In this paper we show how the Parameter Switching algorithm, utilized initially to approximate attractors of a general class of nonlinear dynamical systems, can be utilized also as a synchronization-induced method. Two illustrative examples are considered: the Lorenz system and the Rabinovich-Fabrikant system.

nlin.CD↗
arXiv2015

Looking more closely to the Rabinovich-Fabrikant system

Recently, we look more closely into the Rabinovich-Fabrikant system, after a decade of the study in [Danca & Chen, 2004], discovering some new characteristics such as cycling chaos, transient chaos, chaotic hidden attractors and a new kind of saddles-like attractor. In addition to extensive and accurate numerical analysis, on the assumptive existence of heteroclinic orbits, we provide a few of their approximations.

nlin.CD↗
arXiv2014

Parameter switching in a generalized Duffing system: Finding the stable attractors

This paper presents a simple periodic parameter-switching method which can find any stable limit cycle that can be numerically approximated in a generalized Duffing system. In this method, the initial value problem of the system is numerically integrated and the control parameter is switched periodically within a chosen set of parameter values. The resulted attractor matches with the attractor obtained by using the average of the switched values. The accurate match is verified by phase plots and Hausdorff distance measure in extensive simulations.

nlin.CD↗
arXiv2014

Lyapunov exponents of a class of piecewise continuous systems of fractional order

In this paper, we prove that a class of autonomous piecewise continuous systems of fractional order has well-defined Lyapunov exponents. For this purpose, based on some known results from differential inclusions of integer and fractional order and differential equations with discontinuous right-hand side, the associated discontinuous initial value problem is approximated with a continuous one of fractional order. Then, the Lyapunov exponents are numerically determined using, for example, the known Wolf's algorithm. Three examples of piecewise continuous chaotic systems of fractional order are simulated and analyzed: Sprott's system, Chen's system and Shimizu-Morioka's system.

nlin.CD↗
arXiv2014

Continuous approximations of a class of piece-wise continuous systems

In this paper we provide a rigorous mathematical foundation for continuous approximations of a class of systems with piece-wise continuous functions. By using techniques from the theory of differential inclusions, the underlying piece-wise functions can be locally or globally approximated. The approximation results can be used to model piece-wise continuous-time dynamical systems of integer or fractional-order. In this way, by overcoming the lack of numerical methods for diffrential equations of fractional-order with discontinuous right-hand side, unattainable procedures for systems modeled by this kind of equations, such as chaos control, synchronization, anticontrol and many others, can be easily implemented. Several examples are presented and three comparative applications are studied.

nlin.CD↗
arXiv2014

Synchronization of piece-wise continuous systems of fractional order

The aim of this study is to prove analytically that synchronization of a piece-wise continuous class of systems of fractional order can be achieved. Based on our knowledge, there are no numerical methods to integrate differential equations with discontinuous right hand side of fractional order which model these systems. Therefore, via Filippov's regularization [1] and Cellina's Theorem [2,3], we prove that the initial value problem can be converted into a continuous problem of fractional-order, to which numerical methods for fractional orders apply. In this way, the synchronization problem transforms into a standard problem for continuous systems of fractional order. Three examples of fractional-order piece-wise systems are considered: Sprott system, Chen and Shimizu-Morioka system.

nlin.CD↗
arXiv2011

Synthesizing attractors of Hindmarsh-Rose neuronal systems

In this paper a periodic parameter switching scheme is applied to the Hindmarsh-Rose neuronal system to synthesize certain attractors. Results show numerically, via computer graphic simulations, that the obtained synthesized attractor belongs to the class of all admissible attractors for the Hindmarsh-Rose neuronal system and matches the averaged attractor obtained with the control parameter replaced with the averaged switched parameter values. This feature allows us to imagine that living beings are able to maintain vital behavior while the control parameter switches so that their dynamical behavior is suitable for the given environment.

nlin.CD↗