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In this paper, we carefully reexamine the chaotic RF model, first studied by Rabinovich and Fabrikant [1979], and have found many new and rich complex dynamics of the model that were mostly not reported before. The chaotic RF model has proved to be a great challenge to classical numerical methods, in the sense that most classical numerical methods have not been very successful in the study of complex dynamics of this special RF model. Therefore, in this paper, we present and apply a special numerical method, the Local Iterative Linearization (LIL) method, along with a special Turbo Pascal code based on this accurate LIL algorithm, for a careful numerical study of this complex RF model. Many interesting new findings are summarized and reported in this paper. [ABSTRACT FROM AUTHOR]

The paper discusses the fractional Rössler system and the dependence of its dynamics on some entry parameters. An explicit algorithm for a priori determination of fractional Hopf bifurcations is derived and scenarios documenting a route of the system from stability to chaos are performed with respect to a varying system’s fractional order as well as to a varying system’s coefficient. Contrary to the existing results, the searched values of the fractional Hopf bifurcations follow directly from a revealed analytical dependence between these two systems’ entries. Their various critical values are established and confirmed by numerical experiments demonstrating not only the loss of stability of an equilibrium point, but also other phenomena of transition to chaotic behavior. In addition, we suggest an active control method for synchronization of two chaotic fractional-order Rössler systems. Our theoretical analysis enables to synchronize them for any value of a free parameter under which the master system displays a chaotic behavior. [ABSTRACT FROM AUTHOR]

This paper introduces the capacitated vehicle routing problem with recycling in reverse logistics, and designs a food chain algorithm for it. Some illustrative examples are selected to conduct simulation and comparison. Numerical results show that the performance of the food chain algorithm is better than the genetic algorithm, particle swarm optimization as well as quantum evolutionary algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we present a simple and accessible way to enhance the stable behaviors of a chaotic dynamical system which models a cancer growth, as presented in [Itik & Banks, 2010]. The algorithm presented in [Danca et al., 2012], approximates numerically any attractor of a system belonging to a defined class of dynamical systems, by alternating the control parameter in relatively short periods of time. When switching the control parameter within a set of values corresponding to some chaotic behaviors, the result may be a stable evolution or, reversely a chaotic behavior may be obtained by switching the parameter within a set of values corresponding to stable evolutions. This apparently surprising phenomenon is, in fact, a generalization of the known Parrondo's paradox. [ABSTRACT FROM AUTHOR]

In this paper, we examine the localization problem of compact invariant sets of systems with the differentiable right-side. The localization procedure consists in applying the iterative algorithm based on the first order extremum condition originally proposed by one of authors for periodic orbits. Analysis of a location of compact invariant sets of the Lanford system is realized for all values of its bifurcational parameter. [ABSTRACT FROM AUTHOR]

In this paper, a general autonomous van der Pol-Duffing oscillator is studied. Several issues, such as periodic bifurcations and the dynamical structures of the system are investigated either analytically or numerically. Especially, a phenomenon of hidden attractors is noticed and an algorithm for the location of hidden attractors is given. The obtained results show that hidden attractors exist around chaotic attractors. [ABSTRACT FROM AUTHOR]

Escape time algorithm is a classical algorithm to calculate the Julia sets, but it has the disadvantage of dull color and cannot record the iterative process of the points. In this paper, we present the equipotential point algorithm to calculate the Julia sets by recording the strike frequency of the points in the iterative process. We calculate and analyze the Julia sets in the complex plane by using this algorithm. Finally, we discuss the iteration trajectory of a single point. [ABSTRACT FROM AUTHOR]

Dynamic routing algorithm based on the shortest path principle is criticized due to the oscillation induced by such routing scheme. In the present work, we propose the model of TCP/RED algorithm by a new frame of dynamic routing, based on the measurement of occupation ratio of router buffer for different links, which only requires the information of the queue size at the buffer of the router, to stabilize the system. We classify several types of equilibrium and employ the numerical method to study the stability of the steady state. Our numerical results show that the careful selection of the parameters characterizing the dynamic routing algorithm can stabilize the system in some cases. [ABSTRACT FROM AUTHOR]

Recently, simple dynamical systems such as the 1-d maps on the interval, gained significant attention in the context of statistical physics and complex systems. The decay of correlations in these systems, can be characterized and measured by correlation functions. In the context of symbolic dynamics of the nonchaotic multifractal attractors (i.e. Feigenbaum attractors), one observable, the symbol-to-symbol correlation function, for the generating partition of the logistic map, is rigorously introduced and checked by numerical experiments. Thanks to the Metropolis-Stein-Stein (MSS) algorithm, this observable can be calculated analytically, giving predictions in absolute accordance with numerical computations. The deep, algorithmic structure of the observable is revealed clearly reflecting the complexity of the multifractal attractor. [ABSTRACT FROM AUTHOR]

Asymptotic solutions of parabolic boundary value problems are studied in a neighborhood of both an equilibrium state and a cycle in near-critical cases which can be considered as infinite-dimensional due to small values of the diffusion coefficients. Algorithms are developed to construct normalized equations in such situations. Principle difference between bifurcations in two-dimensional and one-dimensional spatial systems is demonstrated. [ABSTRACT FROM AUTHOR]

We consider the three-parameter numerical continuation of branch points in dynamical systems, with emphasis on the continuation of branch points of periodic orbits. We consider both, the case of branch points along one-parameter families of limit cycles (typical in systems with symmetry), and the case of branch points along fold-curves of limit cycles (typical in generic systems). We discuss new algorithms based on bordered matrices for both detection and continuation. We apply the techniques to a model of a chemical reactor, a model of an electronic circuit, and a model from celestial mechanics. Our algorithms have been implemented in freely available software. [ABSTRACT FROM AUTHOR]

We study multigrid methods in the context of continuation methods for reaction–diffusion systems, where the Bi-CGSTAB and GMRES methods are used as the relaxation scheme for the V-cycle, W-cycle and full approximation schemes, respectively. In particular, we apply the results of Brown and Walker [1997] to investigate how the GMRES method can be used to solve nearly singular systems that occur in continuation problems. We show that for the sake of switching branches safely, one would rather solve a perturbed problem near bifurcation points. We propose several multigrid-continuation algorithms for curve-tracking in nonlinear elliptic eigenvalue problems. Our numerical results show that the algorithms proposed have the advantage of being robust and easy to implement. [ABSTRACT FROM AUTHOR]