arXiv subjects

Marius-F. Danca

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

At least 19 recordsLinked to original sources

arXiv2026

Improved Matlab code for Lyapunov exponents of fractional order systems

This paper presents an improved Matlab routine, FO_LE, for the numerical computation of Lyapunov exponents of fractional-order systems modeled by Caputo's derivative. It is conceived as an enhanced version of the former FO_Lyapunov and FO_NC_Lyapunov codes for commensurate and non-commensurate orders, respectively. The proposed approach replaces the Gram-Schmidt orthogonalization procedure with QR-based reorthonormalization and uses the new quadratic LIL predictor-corrector scheme for the integration of the extended variational system. Compared with the former implementations, the present routine benefits from the higher order of the fractional integrator LIL and applies to both commensurate and non-commensurate models. Like the previous code, FO_LE retains the full memory structure of the underlying Caputo model. The Matlab code for the LIL solver and for the computation of Lyapunov exponents with FO_LE are provided, while a fast implementation of LIL for commensurate and non-commensurate orders, LIL_nc, is available on MathWorks File Exchange. A benchmark problem with exact solution is used to compare the LIL-based solver with ABM-type methods, whereas the Rabinovich-Fabrikant system illustrates the computation of Lyapunov exponents in different dynamical regimes. The results indicate that the proposed implementation is a compact, robust, and efficient tool for the numerical study of stability and chaos in fractional-order systems.

nlin.CD↗

arXiv2024

Memory principle of the Matlab code for Lyapunov Exponents of fractional order

The paper presents two representative classes of Impulsive Fractional Differential Equations defined with generalized Caputoś derivative, with fixed lower limit and changing lower limit, respectively. Memory principle is studied and numerical examples are considered. The problem of the memory principle of the Matlab code for Lyapunov exponents of fractional order systems [Danca & Kuznetsov, 2018] is analyzed.

nlin.CD↗

arXiv2024

Approximation and decomposition of attractors of a Hopfield neural network system

In this paper, the Parameter Switching (PS) algorithm is used to approximate numerically attractors of a Hopfield Neural Network (HNN) system. The PS algorithm is a convergent scheme designed for approximating attractors of an autonomous nonlinear system, depending linearly on a real parameter. Aided by the PS algorithm, it is shown that every attractor of the HNN system can be expressed as a convex combination of other attractors. The HNN system can easily be written in the form of a linear parameter dependence system, to which the PS algorithm can be applied. This work suggests the possibility to use the PS algorithm as a control-like or anticontrol-like method for chaos.

nlin.CD↗

arXiv2023

Mandelbrot set as a particular Julia set of Fractional Order, echipotential lines and external rays of Mandelbrot and Julia sets of Fractional Order

In this paper it is shown analytically and computationally that the Mandelbrot set of integer order are particular cases of Julia sets of Caputo s like fractional order. Also the differences between the fractional-order Mandelbrot set and Julia sets from their integer-order variants are revealed. Equipotential lines and external rays of Mandelbrot set and Julia sets of fractional order are determined.

nlin.CD↗

arXiv2023

Chaotic hidden attractor in a fractional order system modelling the interaction between dark matter and dark energy

In this paper the dynamics of a fractional order system modelling the interaction between dark matter and dark energy is analytically and numerically studied. It is shown for the first time that systems modelling the interaction between dark matter and dark energy, chaotic hidden attractors can be present. The chaotic attractor co-exists with two asymptotically stable equilibria. Equilibria of the linearized system exhibit a center-like behavior. The numerical integration is done by means of the Adams-Bashforth-Moulton scheme and the finite Lyapunov exponents are numerically determined with a dedicated Matlab code. The 3D representation of the chaotic hidden attractor reveals the fact it is not connected with the equilibria, being `hidden` somewhere in the considered phase space.

math.DS↗

arXiv2023

On the stability domain of a class of linear systems of fractional order

In this paper, the shape of the stability domain S^q for a class of difference systems defined by the Caputo forward difference operator D^q of order q\in (0, 1) is numerically analyzed. It is shown numerically that due to of power of the negative base in the expression of the stability domain, in addition to the known cardioid-like shapes, S^q could present supplementary regions where the stability is not verified. The Mandelbrot map of fractional order is considered as an illustrative example. In addition, it is conjectured that for $q < 0.5$, the shape of S^q does not cover the main body of the underlying Mandelbrot set of fractional order as in the case of integer order.

math.DS↗

arXiv2022

Mandelbrot set and Julia sets of fractional order

In this paper the fractional-order Mandelbrot and Julia sets in the sense of $q$-th Caputo-like discrete fractional differences, for $q\in(0,1)$, are introduced and several properties are analytically and numerically studied. Some intriguing properties of the fractional models are revealed. Thus, for $q\uparrow1$, contrary to expectations, it is not obtained the known shape of the Mandelbrot of integer order, but for $q\downarrow0$. Also, we conjecture that for $q\downarrow0$, the fractional-order Mandelbrot set is similar to the integer-order Mandelbrot set, while for $q\downarrow0$ and $c=0$, one of the underlying fractional-order Julia sets is similar to the integer-order Mandelbrot set. In support of our conjecture, several extensive numerical experiments were done. To draw the Mandelbrot and Julia sets of fractional order, the numerical integral of the underlying initial values problem of fractional order is used, while to draw the sets, the escape-time algorithm adapted for the fractional-order case is used. The algorithm is presented as pseudocode.

nlin.CD↗

arXiv2022

Symmetry-breaking and bifurcation diagrams of fractional-order maps

In this paper two important aspects related to Caputo fractional-order discrete variant of a class of maps defined on the complex plane, are analytically and numerically revealed: attractors symmetry-broken induced by the fractional-order and the sensible problem of determining the right bifurcation diagram of discrete systems of fractional-order. It is proved that maps of integer order with dihedral symmetry or cycle symmetry loose their symmetry once they are transformed in fractional-order maps. Also, it is conjectured that, contrarily to integer-order maps, determining the bifurcation diagrams of fractional-order maps is far from being a clarified problem. Two examples are considered: dihedral logistic map and cyclic logistic map.

math.DS↗

arXiv2022

Controlling dynamics of a COVID--19 mathematical model using a parameter switching algorithm

In this paper the dynamics of an autonomous mathematical models of COVID-19 depending on a real parameter bifurcation, is controlled by switching periodically the parameter value. For this purpose the Parameter Switching (PS) algorithm is used. With this technique, it is proved that every attractor of the considered system can be numerically approximated and, therefore, the system can be determined to evolve along, e.g., a stable periodic motion or a chaotic attractor. In this way, the algorithm can be considered as a chaos control or anticontrol (chaoticization)-like algorithm. Contrarily to existing chaos control techniques which generate modified attractors, the obtained attractors with the PS algorithm belong to the set system attractors. It is analytically shown that using the PS algorithm, every system attractor can be expressed as a convex combination of some existing attractors. Moreover, is proved that the PS algorithm can be viewed as a generalization of Parrondo\textsc{\char13}s paradox.

math.DS↗

arXiv2022

Fractional order logistic map; Numerical approach

In this paper the fractional order logistic map in the sense of Caputo's fractional differences is numerically approached. It is shown that the necessary iterations number to avoid transients must be of order of thousand, not of order of hundreds as commonly used in several works. Also, it is revealed an interesting phenomenon according to which for every initial condition it correspond a different bifurcation diagram. This phenomenon seems to appear also in other FO difference systems, fact which could represent an obstacle for the numerical analysis. A short Matlab code is used to obtain the results.

math.NA↗

arXiv2021

Hopfield Neuronal Network of Fractional Order: A note on its numerical integration

In this paper, the commensurate fractional-order variant of an Hopfield neuronal network is analyzed. The system is integrated with the ABM method for fractional-order equations. Beside the standard stability analysis of equilibria, the divergence of fractional order is proposed to determine the instability of the equilibria. The bifurcation diagrams versus the fractional order, and versus one parameter, reveal a strange phenomenon suggesting that the bifurcation branches generated by initial conditions outside neighborhoods of unstable equilibria are spurious sets although they look similar with those generated by initial conditions close to the equilibria. These spurious sets look ``delayed'' in the considered bifurcation scenario. Once the integration step-size is reduced, the spurious branches maintain their shapes but tend to the branches obtained from initial condition within neighborhoods of equilibria. While the spurious branches move once the integration step size reduces, the branches generated by the initial conditions near the equilibria maintain their positions in the considered bifurcation space. This phenomenon does not depend on the integration-time interval, and repeats in the parameter bifurcation space.

math.NA↗

arXiv2021

Matlab code for Lyapunov exponents of fractional-order systems, Part II: The non-commensurate case

In this paper the Benettin-Wolf algorithm to determine all Lyapunov exponents adapted to a class of non-commensurate fractional-order systems modeled by Caputo's derivative and the corresponding Matlab code are presented. The paper continues the work started in [Danca & Kuznetsov, 2018], where the Matlab code of commensurate fractional-order systems is given. To integrate the extended system, the Adams-Bashforth-Moulton for fractional differential equations is utilized. As the the Matlab program for commensurate-order systems, the program presented in this paper prints and plots all Lyapunov exponents as function of time. The program can be simply adapted to plot the evolution of the Lyapunov exponents as function of orders, or a function of a bifurcation parameter. A special attention is payed to the periodicity of fractional-order systems and its influences. The case of the Lorenz system is considered.

physics.comp-ph↗

arXiv2020

Coexisting Hidden and self-excited attractors in an economic system of integer or fractional order

In this paper the dynamics of an economic system with foreign financing, of integer or fractional order, are analyzed. The symmetry of the system determines the existence of two pairs of coexisting attractors. The integer-order version of the system proves to have several combinations of coexisting hidden attractors with self-excited attractors. Because one of the system variables represents the foreign capital in ow, the presence of hidden attractors could be of a real interest in economic models. The fractional-order variant presents another interesting coexistence of attractors in the fractional order space.

q-fin.GN↗

arXiv2020

Hidden and self-exited attractors in a heterogeneous Cournot oligopoly model

In this paper it is numerically proved that a heterogeneous Cournot oligopoly model presents hidden and self-excited attractors. The system has a single equilibrium and a line of equilibria. The bifurcation diagrams show that the system admits several attractors coexistence windows, where the hidden attractors can be found. Intensive numerical tests have been done.

nlin.CD↗

arXiv2019

Rich dynamics and anticontrol of extinction in a prey-predator system

This paper reveals some new and rich dynamics of a two-dimensional prey-predator system and to anticontrol the extinction of one of the species. For a particular value of the bifurcation parameter, one of the system variable dynamics is going to extinct, while another remains chaotic. To prevent the extinction, a simple anticontrol algorithm is applied so that the system orbits can escape from the vanishing trap. As the bifurcation parameter increases, the system presents quasiperiodic, stable, chaotic and also hyperchaotic orbits. Some of the chaotic attractors are Kaplan-Yorke type, in the sense that the sum of its Lyapunov exponents is positive. Also, atypically for undriven discrete systems, it is numerically found that, for some small parameter ranges, the system seemingly presents strange nonchaotic attractors. It is shown both analytically and by numerical simulations that the original system and the anticontrolled system undergo several Neimark-Sacker bifurcations. Beside the classical numerical tools for analyzing chaotic systems, such as phase portraits, time series and power spectral density, the 0-1 test is used to differentiate regular attractors from chaotic attractors.

nlin.CD↗

arXiv2019

Chaos control in the fractional order logistic map via impulses

In this paper the chaos control in the discrete logistic map of fractional order is obtained with an impulsive control algorithm. The underlying discrete initial value problem of fractional order is considered in terms of Caputo delta fractional difference. Every $Δ$ steps, the state variable is instantly modified with the same impulse value, chosen from a bifurcation diagram versus impulse. It is shown that the solution of the impulsive control is bounded. The numerical results are verified via time series, histograms, and the 0-1 test. Several examples are considered

nlin.CD↗

arXiv2019

Chaos suppression in a Gompertz-like discrete system of fractional order

In this paper we introduce the fractional-order variant of a Gompertz-like discrete system. The chaotic behavior is suppressed with an impulsive control algorithm. The numerical integration and the Lyapunov exponent are obtained by means of the discrete fractional calculus. To verify numerically the obtained results, beside the Lyapunov exponent, the tools offered by the 0-1 test are used.

math.DS↗

arXiv2018

Approximating hidden chaotic attractors via parameter switching

In this paper, the problem of approximating hidden chaotic attractors of a general class of nonlinear systems is investigated. The Parameter Switching (PS) algorithm is utilized, which switches the control parameter within a given set of values with the initial value problem numerically solved. The PS-generated generated attractor approximates the attractor obtained by averaging the control parameter with the switched values, which represents the hidden chaotic attractor. The hidden chaotic attractors of a generalized Lorenz system and the Rabinovich-Fabrikant system are simulated for illustration

nlin.CD↗