Paulo C. Rech

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

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Publications (45)58.61 Total impact

  • Cristiane Stegemann, Paulo C. Rech
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    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). · 0.92 Impact Factor
  • Paulo C. Rech
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    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. · 1.63 Impact Factor
  • Willian T Prants, Paulo C Rech
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    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. · 1.03 Impact Factor
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    Amanda C. Mathias, Paulo C. Rech
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    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. · 0.92 Impact Factor
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    Amanda C Mathias, Paulo C Rech
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    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.
    Chaos (Woodbury, N.Y.) 12/2012; 22(4):043147. · 1.80 Impact Factor
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    Amanda C Mathias, Paulo C Rech
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    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. · 1.88 Impact Factor
  • Gabriela A. Casas, Paulo C. Rech
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    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. · 2.77 Impact Factor
  • Marcos J. Correia, Paulo C. Rech
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    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. · 1.35 Impact Factor
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    Paulo C. Rech
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    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 01/2012; 29(6). · 0.92 Impact Factor
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    ABSTRACT: We report numerical results on the existence of periodic structures embedded in chaotic and hyperchaotic regions on the Lyapunov exponent diagrams of a 4-dimensional Chua system. The model was obtained from the 3-dimensional Chua system by the introduction of a feedback controller. Both the largest and the second largest Lyapunov exponents were considered in our colorful Lyapunov exponent diagrams, and allowed us to characterize periodic structures and regions of chaos and hyperchaos. The shrimp-shaped periodic structures appear to be malformed on some of Lyapunov exponent diagrams, and they present two different bifurcation scenarios to chaos when passing the boundaries of itself, namely via period-doubling and crisis. Hyperchaos-chaos transition can also be observed on the Lyapunov exponent diagrams for the second largest exponent.
    Chaos (Woodbury, N.Y.) 09/2011; 21(3):033105. · 1.80 Impact Factor
  • Marcos J Correia, Paulo C Rech
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    ABSTRACT: In this paper we report a new four-dimensional autonomous system, constructed from a Lorenz system by introducing an adequate feedback controller to the third equation. We use a numerical method that considers the second largest Lyapunov exponent value as a measure of hyperchaotic motion, to construct a two-dimensional parameter-space color plot for this system. Different levels of hyperchaos are represented in this plot by a continuously changing yellow-red scale. Practical applications of this plot includes, by instance, walking in the parameter-space of hyperchaotic systems along suitable paths.
    Journal of Physics Conference Series 04/2011; 285(1):012017.
  • Gabriela A Casas, Paulo C Rech
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    ABSTRACT: In this work we consider a situation in which the two parameters of a Hénon map are linearly modulated by the output of another Hénon map, whose parameters are constant in time but can be adjusted. More specifically, here we numerically investigate modifications in basins of attraction of coexisting states, and shift in the location of critical points in bifurcation diagrams of a Hénon map, by virtue of periodic parametric modulation from another Hénon map.
    Journal of Physics Conference Series 04/2011; 285(1):012001.
  • Paulo C. Rech
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    ABSTRACT: In this paper we investigate numerically the parameter-space of an autonomous system of four nonlinear first-order ordinary differential equations, which represents a Hopfield neural network with four neurons. The study considers three independent two-dimensional cross-sections of the three-dimensional parameter-space generated by this mathematical model, every constructed considering Lyapunov exponent values. We show that is possible to completely characterize the dynamics of the system based in these three plots, which are representative of the three-dimensional parameter-space as a whole.
    Neurocomputing. 01/2011; 74:3361-3364.
  • Paulo C. Rech
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    ABSTRACT: We report some two-dimensional parameter-space diagrams numerically obtained for the multi-parameter Hindmarsh-Rose neuron model. Several different parameter planes are considered, and we show that regardless of the combination of parameters, a typical scenario is preserved: for all choice of two parameters, the parameter-space presents a comb-shaped chaotic region immersed in a large periodic region. We also show that exist regions close these chaotic region, separated by the comb teeth, organized themselves in period-adding bifurcation cascades.
    Physics Letters A 01/2011; 375:1461-1464. · 1.63 Impact Factor
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    Eduardo L. Brugnago, Paulo C. Rech
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    ABSTRACT: We study a pair of nonlinearly coupled identical chaotic sine square maps. More specifically, we investigate the chaos suppression associated with the variation of two parameters. Two-dimensional parameter-space regions where the chaotic dynamics of the individual chaotic sine square map is driven towards regular dynamics are delimited. Additionally, the dynamics of the coupled system is numerically characterized as the parameters are changed.
    Chinese Physics Letters 01/2011; 28(11). · 0.92 Impact Factor
  • P C Rech, M W Beims
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    ABSTRACT: The stability of periodic orbits under translation and also dilatation (or contraction) of coordinates is analysed using a q-Jacobian. Specifically we analyse the stability of period-1 and period-2 orbits of the quadratic map and show that the dilatation changes the parameter values for which bifurcations take place. The dilatation (|q| > 1) or contraction (|q| < 1) can be used to stabilize (destabilize) the unstable (stable) periodic orbits. In addition, dilatation effects on the stability depend only on the period of the orbit, and not on the location of orbital points in phase-space. This is shown to be true for any period-n orbit of the quadratic map. As a possible practical application, we suggest that the dilatation (contraction) considered here could represent temperature variations in physical systems.
    Journal of Physics Conference Series 09/2010; 246(1):012006.
  • Holokx A. Albuquerque, Paulo C. Rech
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    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 06/2010; 40(2):189 - 194. · 1.29 Impact Factor
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    ABSTRACT: In this paper we investigate three two-dimensional parameter spaces of a three-parameter set of autonomous differential equations used to model the Chua oscillator, where the piecewise-linear function usually taken to describe the nonlinearity of the Chua diode has been replaced by a cubic polynomial. It is made by using three independent two-dimensional cross sections of the three-dimensional parameter space generated by the model, which contains three parameters. We show that, independent of the parameter set considered in plots, all the diagrams present periodic structures embedded in a large chaotic region, and we also show that these structures organize themselves in period-adding cascades. We argue that these selected two-dimensional cross sections can be representative of the three-dimensional parameter space as a whole, in the range of parameters here investigated.
    Chaos (Woodbury, N.Y.) 06/2010; 20(2):023103. · 1.80 Impact Factor
  • Marcos J. Correia, Paulo C. Rech
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    ABSTRACT: In this letter we report a new four-dimensional autonomous system, constructed from a chaotic Lorenz system by introducing an adequate feedback controller to the third equation. We show that when parameters are conveniently chosen, the control method can drive the chaotic Lorenz system to hyperchaotic regions. Analytical and numerical procedures are conducted to study the dynamical behaviors of the proposed new system.
    International Journal of Bifurcation and Chaos 01/2010; 20:3295-3301. · 0.92 Impact Factor
  • João C. Xavier, Paulo C. Rech
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    ABSTRACT: We analytically investigate the dynamics of the generalized Lorenz equations obtained by Stenflo for acoustic gravity waves. By using Descartes' Rule of Signs and Routh–Hurwitz Test, we decide on the stability of the fixed points of the Lorenz–Stenflo system, although without explicit solution of the eigenvalue equation. We determine the precise location where pitchfork and Hopf bifurcation of fixed points occur, as a function of the parameters of the system. Parameter-space plots, Lyapunov exponents, and bifurcation diagrams are used to numerically characterize periodic and chaotic attractors.
    International Journal of Bifurcation and Chaos 01/2010; 20:145-152. · 0.92 Impact Factor