Paulo C. Rech

Universidade do Estado de Santa Catarina, Joinville, Santa Catarina, Brazil

Are you Paulo C. Rech?

Claim your profile

Publications (55)75.11 Total impact

  • Rodrigo A. da Silva · Paulo C. Rech
    [Show abstract] [Hide abstract]
    ABSTRACT: We investigate a parameter plane of a set of three autonomous, ten-parameter, first-order nonlinear ordinary differential equations, which models a tri-trophic food web system. By using Lyapunov exponents, bifurcation diagrams, and trajectories in the phase-space, to numerically characterize the dynamics of the model in a parameter plane, we show that it presents typical periodic structures embedded in a chaotic region, forming a spiral structure that coils up around a focal point while period-adding bifurcations take place.
    Applied Mathematics and Computation 03/2015; 254. DOI:10.1016/j.amc.2014.12.115 · 1.60 Impact Factor
  • Paulo C. Rech
    [Show abstract] [Hide abstract]
    ABSTRACT: Different ways to numerically characterize hyperchaotic regions in parameter planes of a 5D continuous-time nonlinear dynamical system are utilized in this work. The method considers the three largest Lyapunov exponents, to construct two-dimensional parameter planes colorful plots. In some these plots different levels of hyperchaos are represented by a continuously changing yellow to red scale, while in other different colors mean different number of positive Lyapunov exponents.
    Applied Mathematics and Computation 11/2014; 247:13–17. DOI:10.1016/j.amc.2014.08.084 · 1.60 Impact Factor
  • Fabiola G. Prants · Paulo C. Rech
    [Show abstract] [Hide abstract]
    ABSTRACT: We report on the dynamics in a parameter plane of a continuous-time damped system driven by a periodic forcing. The dynamics is characterized by considering the Lyapunov exponents spectrum and conventional bifurcation diagrams, to discriminate periodic, quasiperiodic, and chaotic behaviors for each point in this parameter plane, according two parameters are simultaneously varied. Periodic structures born in a quasiperiodic region and embedded in a chaotic region, the so-called Arnold tongues, are observed. We show that the Arnold tongues periodic distribution is highly organized in a mixed set of two period-adding sequences. Other three typical periodic structures born and embedded in a chaotic region were observed, also individually organized in a mixed set of two period-adding sequences.
    Physics of Condensed Matter 09/2014; 87(9). DOI:10.1140/epjb/e2014-50368-9 · 1.46 Impact Factor
  • Paulo C. Rech
    [Show abstract] [Hide abstract]
    ABSTRACT: We investigate periodicity suppression in two-dimensional parameter-spaces of discrete-and continuous-time nonlinear dynamical systems, modeled respectively by a two-dimensional map and a set of three first-order ordinary differential equations. We show for both cases that, by varying the amplitude of an external periodic forcing with a fixed angular frequency, windows of periodicity embedded in a chaotic region may be totally suppressed.
    International Journal of Bifurcation and Chaos 07/2014; 24(07):1430020. DOI:10.1142/S0218127414300201 · 1.02 Impact Factor
  • Cristiane Stegemann · Paulo C. Rech
    [Show abstract] [Hide abstract]
    ABSTRACT: We report results of a numerical investigation on a two-dimensional cross-section of the parameter-space of a set of three autonomous, eight-parameter, first-order ordinary differential equations, which models tumor growth. The model considers interaction between tumor cells, healthy tissue cells, and activated immune system cells. By using Lyapunov exponents to characterize the dynamics of the model in a particular parameter plane, we show that it presents typical self-organized periodic structures embedded in a chaotic region, that were before detected in other models. We show that these structures organize themselves in two independent ways: (i) as spirals that coil up toward a focal point while undergoing period-adding bifurcations and, (ii) as a sequence with a well-defined law of formation, constituted by two mixed period-adding bifurcation cascades.
    International Journal of Bifurcation and Chaos 01/2014; 24(02). DOI:10.1142/S0218127414500230 · 1.02 Impact Factor
  • Rodrigo A. da Silva · Paulo C. Rech
    [Show abstract] [Hide abstract]
    ABSTRACT: We investigate analytically and numerically the dynamics of the Rikitake system. The Routh-Hurwitz criterion is used to study the stability of the equilibrium points of the differential equation system model, as functions of two parameters. The dynamics of the model are numerically studied using diagrams that associate colors to the largest Lyapunov exponent value, in two-dimensional parameter spaces. Additionally, phase-space plots and bifurcation diagrams are used to distinguish periodic and chaotic attractors.
    Chinese Physics Letters 12/2013; 30(12):120501. DOI:10.1088/0256-307X/30/12/120501 · 0.95 Impact Factor
  • Source
    Paulo C. Rech
    [Show abstract] [Hide abstract]
    ABSTRACT: We investigate changes in periodicity, and even its suppression, by external periodic forcing in different two-dimensional maps, namely the Hénon map and the sine square map. By varying the amplitude of a periodic forcing with a fixed angular frequency, we show through numerical simulations in parameter-spaces that changes in periodicity may take place. We also show that windows of periodicity embedded in a chaotic region may be totally suppressed.
    Physics Letters A 10/2013; 377(s 31–33):1881–1884. DOI:10.1016/j.physleta.2013.05.049 · 1.63 Impact Factor
  • Antonio Endler · Paulo C. Rech
    [Show abstract] [Hide abstract]
    ABSTRACT: A transition from Mandelbrot-like sets to Arnold tongues is characterized via a coupling of two non-identical quadratic maps proposed by us. A two-dimensional parameter-space considering the parameters of the individual quadratic maps was used to demonstrate numerically the event. The location of the parameter sets where Naimark-Sacker bifurcations occur, which is exactly the place where Arnold tongues of arbitrary periods are born, was computed analytically.
    Applied Mathematics and Computation 10/2013; 222:559-563. DOI:10.1016/j.amc.2013.08.001 · 1.60 Impact Factor
  • Paulo C. Rech
    [Show abstract] [Hide abstract]
    ABSTRACT: Three two-dimensional parameter planes of a three-parameter, three-dimensional set of autonomous nonlinear first-order differential equations used to model the A2 symmetric flow are investigated. This is done by using the three two-dimensional cross sections of the three-dimensional parameter-space generated by the model. We show that regardless of the two-parameter set considered in the parameter plane plots, all the diagrams present periodic structures embedded in a large chaotic region. We also show that these periodic structures organize themselves in different ways, including sequences whose periods have a well-defined law of formation that can be written in a closed form, and sequences organized in period-adding bifurcation cascades.
    Applied Mathematics and Computation 09/2013; 220:208-212. DOI:10.1016/j.amc.2013.06.044 · 1.60 Impact Factor
  • Paulo C. Rech
    [Show abstract] [Hide abstract]
    ABSTRACT: We investigate the dynamical behavior of a symmetric linear coupling of three quadratic maps with exponential terms, and identify various interesting features as a function of two control parameters. In particular, we investigate the emergence of quasiperiodic states arising from Naimark—Sacker bifurcations of stable period-1, period-2, and period-3 orbits. We also investigate the multistability in the same coupling. Lyapunov exponents, parameter planes, phase space portraits, and bifurcation diagrams are used to investigate transitions from periodic to quasiperiodic states, from quasiperiodic to mode-locked states and to chaotic states, and from chaotic to hyperchaotic states.
    Chinese Physics B 08/2013; 22(8):080202. DOI:10.1088/1674-1056/22/8/080202 · 1.39 Impact Factor
  • Paulo C. Rech
    [Show abstract] [Hide abstract]
    ABSTRACT: Parameter plane plots related to a periodically forced compound Korteweg-de Vries-Burgers system, which is modeled by a third-order partial differential equation, are reported. It is shown that typical periodic structures embedded in a chaotic region in these parameter planes, organize themselves in different ways. There are bifurcation sequences whose periods have a well-defined law of formation, that may be written in a closed form, and there are bifurcation sequences self-organized in period-adding cascades.
    Physics of Condensed Matter 08/2013; 86(8). DOI:10.1140/epjb/e2013-40238-5 · 1.46 Impact Factor
  • Willian T Prants · Paulo C Rech
    [Show abstract] [Hide abstract]
    ABSTRACT: We report the results of our numerical investigation on a parameter plane of a set of six autonomous five-parameter first-order ordinary differential equations, namely a coupling of two identical chaotic Rössler oscillators. Using the Lyapunov exponents spectrum to characterize the dynamics of the model in a parameter plane that considers the asymmetry and the strength of the coupling, we show that this particular parameter plane presents malformed shrimp-shaped periodic structures embedded in a hyperchaotic region, which are self-organized in period-adding bifurcation cascades.
    Physica Scripta 06/2013; 88(1):015001. DOI:10.1088/0031-8949/88/01/015001 · 1.30 Impact Factor
  • Source
    Amanda C. Mathias · Paulo C. Rech
    [Show abstract] [Hide abstract]
    ABSTRACT: We report some results indicating changes in the observed dynamics of the Rössler model under the influence of external sinusoidal forcing. By varying the control parameters of the external sinusoidal forcing, namely the amplitude and the angular frequency, we show, through numerical simulations which include parameter planes and Lyapunov exponents, that the external forcing can produce both chaos-order and order-chaos transitions. We also show that the sinusoidal forcing may generate hyperchaos.
    Chinese Physics Letters 03/2013; 30(3):030502. DOI:10.1088/0256-307X/30/3/030502 · 0.95 Impact Factor
  • Paulo C. Rech
    [Show abstract] [Hide abstract]
    ABSTRACT: This work reports two-dimensional parameter space plots, concerned with a three-dimensional Hopfield-type neural network with a hyperbolic tangent as the activation function. It shows that typical periodic structures embedded in a chaotic region, called shrimps, organize themselves in two independent ways: (i) as spirals that individually coil up toward a focal point while undergo period-adding bifurcations and, (ii) as a sequence with a well-defined law of formation, constituted by two different period-adding sequences inserted between.
    02/2013; 6(1). DOI:10.1007/s13042-013-0222-0
  • Source
    Amanda C Mathias · Paulo C Rech
    [Show abstract] [Hide abstract]
    ABSTRACT: We investigate periodicity suppression by an external periodic forcing in different flows, each modeled by a set of three autonomous nonlinear first-order ordinary differential equations. By varying the amplitude of a sinusoidal forcing with a fixed angular frequency, we show through numerical simulations, including parameter planes plots, phase-space portraits, and the largest Lyapunov exponent, that windows of periodicity embedded in chaotic regions may be totally suppressed. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4772968]
    Chaos (Woodbury, N.Y.) 12/2012; 22(4):043147. DOI:10.1063/1.4772968 · 1.76 Impact Factor
  • Source
    Amanda C Mathias · Paulo C Rech
    [Show abstract] [Hide abstract]
    ABSTRACT: This paper reports two-dimensional parameter-space plots for both, the hyperbolic tangent and the piecewise-linear neuron activation functions of a three-dimensional Hopfield neural network. The plots obtained using both neuron activation functions are compared, and we show that similar features are present on them. The occurrence of self-organized periodic structures embedded in chaotic regions is verified for the two cases.
    Neural networks: the official journal of the International Neural Network Society 07/2012; 34:42-5. DOI:10.1016/j.neunet.2012.06.006 · 2.08 Impact Factor
  • Gabriela A. Casas · Paulo C. Rech
    [Show abstract] [Hide abstract]
    ABSTRACT: We consider a situation in which the two parameters of a Hénon map are modulated by the output of another Hénon map. Two cases are considered. Firstly, we investigate the behavior of the Hénon map when its parameters are modulated by another Hénon map, this last working in a high dissipative regime. Secondly, we use a Hénon map working in a low dissipative regime as the modulation. We show that, regardless of the considered case, multistability can be suppressed by the modulation.Highlights► We consider the modulation of the parameters of a Hénon map, by a second Hénon map. ► We consider the modulator Hénon map working in a high dissipative regime. ► The modulator Hénon map working in a low dissipative regime also is considered. ► Regardless the modulator regime, multistability may be suppressed in the Hénon map.
    Communications in Nonlinear Science and Numerical Simulation 06/2012; 17(6):2570-2578. DOI:10.1016/j.cnsns.2011.10.031 · 2.57 Impact Factor
  • Source
    Paulo C. Rech
    [Show abstract] [Hide abstract]
    ABSTRACT: Some two-dimensional parameter-space diagrams are numerically obtained by considering the largest Lyapunov exponent for a four-dimensional thirteen-parameter Hindmarsh—Rose neuron model. Several different parameter planes are considered, and it is shown that depending on the combination of parameters, a typical scenario can be preserved: for some choice of two parameters, the parameter plane presents a comb-shaped chaotic region embedded in a large periodic region. It is also shown that there exist regions close to these comb-shaped chaotic regions, separated by the comb teeth, organizing themselves in period-adding bifurcation cascades.
    Chinese Physics Letters 06/2012; 29(6). DOI:10.1088/0256-307X/29/6/060506 · 0.95 Impact Factor
  • Marcos J. Correia · Paulo C. Rech
    [Show abstract] [Hide abstract]
    ABSTRACT: In this paper we propose a numerical method to characterize hyperchaotic points in the parameter-space of continuous-time dynamical systems. The method considers the second largest Lyapunov exponent value as a measure of hyperchaotic motion, to construct two-dimensional parameter-space color plots. Different levels of hyperchaos in these plots are represented by a continuously changing yellow–red scale. As an example, a particular system modeled by a set of four nonlinear autonomous first-order ordinary differential equations is considered. Practical applications of these plots include, by instance, walking in the parameter-space of hyperchaotic systems along desirable paths.
    Applied Mathematics and Computation 02/2012; 218(12):6711–6715. DOI:10.1016/j.amc.2011.12.035 · 1.60 Impact Factor
  • Holokx A. Albuquerque · Paulo C. Rech
    [Show abstract] [Hide abstract]
    ABSTRACT: In this letter we investigate, via numerical simulations, the parameter-space of the set of autonomous first-order differential equations of a Chua circuit. We show that this parameter-space presents self-organized periodic structures immersed in a chaotic region, forming a single spiral structure that coils up around a focal point. Additionally, bifurcation diagrams are used to show that those periodic structures also organize themselves in period-adding cascades, along specific directions that point towards this same focal point. Copyright © 2010 John Wiley & Sons, Ltd.
    International Journal of Circuit Theory and Applications 02/2012; 40(2):189 - 194. DOI:10.1002/cta.713 · 1.21 Impact Factor