International Journal of Bifurcation and Chaos

Historically cognition was understood as the result of processes occurring solely in the brain. Recently, however, cognitive scientists and philosophers studying "embodied" or "situated" cognition have begun emphasizing the role of the body and environment in which brains are situated, i.e. they view the brain as an "open system". However, these theorists frequently rely on dynamical systems which are traditionally viewed as closed systems. We address this tension by extending the framework of dynamical systems theory. We show how structures which appear in the state space of an embodied agent differ from those that appear in closed systems, and we show how these structures can be used to model representational processes in embodied agents. We focus on neural networks as models of embodied cognition.

Chaos was historically discovered with the Lorenz attractor in 1963. Several artworks, created before that date, by post-war artists such as Fontana, Crippa, Mathieu and Tobey, seem to anticipate forms and shapes typically shown by chaotic systems. In this paper, we juxtapose these masterpieces with some strange attractors of dynamical system theory in a timeline leading to the question whether artists discovered chaos before scientists.

The aim of this work is twofold — on the one hand, to perform a theoretical analysis of the global behavior organized by a T-point–Hopf in ℤ₂-symmetric systems; on the other hand, to apply the obtained results for a numerical study of Chua's equation, where for the first time this bifurcation is considered.

In a parameterized three-dimensional system of autonomous differential equations, a T-point is a point of the parameter space where a special kind of codimension-two heteroclinic cycle occurs. A more degenerate scenario appears when one of the equilibria involved in such a cycle undergoes a Hopf bifurcation. This degeneration, which corresponds to a codimension-three bifurcation, is called T-point–Hopf and has been recently studied for a generic system. However, the presence of ℤ₂-symmetry may lead to the existence of a double T-point–Hopf heteroclinic cycle, which is responsible for the appearance of interesting global behavior that we will study in this paper.

The theoretical models proposed for two different situations are based on the construction of a Poincaré map. The existence of certain kinds of homoclinic and heteroclinic connections between equilibria and/or periodic orbits is proved and their organization close to the T-point–Hopf bifurcation is described. The numerical phenomena found in Chua's equation strongly agree with the results deduced from the models.

Given an m-parameterized family of n-dimensional vector fields, such that: (i) for some value of the parameters, the family has an equilibrium point, (ii) its linearization has a double zero eigenvalue and no other eigenvalue on the imaginary axis, sufficient conditions on the vector field are given such that the dynamics on the two-dimensional center manifold is locally topologically equivalent to the versal deformation of the planar Takens–Bogdanov bifurcation.

A reaction–diffusion Gierer–Meinhardt model of morphogenesis subject to Dirichlet fixed boundary condition in the one-dimensional spatial domain is considered. We perform a detailed Hopf bifurcation analysis and steady state bifurcation analysis to the system. Our results suggest the existence of spatially nonhomogenous periodic orbits and nonconstant positive steady state solutions, which imply the possibility of complex spatiotemporal patterns of the system. Numerical simulations are carried out to support our theoretical analysis.

Bifurcation tailoring is a method developed to design control laws modifying the bifurcation diagram of a nonlinear dynamical system to a desired one. In its original formulation, this method does not account for the possible presence of constraints on state and/or manipulated inputs. In this paper, a novel formulation of the bifurcation tailoring method overcoming this limitation is presented. In accordance with the proposed approach, a feedforward control law generating an optimal bifurcation diagram is computed by constrained minimization of an objective functional. Then, a feedback control system enforcing stability of the computed equilibrium branch is designed. In this context, bifurcation analysis is exploited to select feedback controller parameters ensuring desired output behavior and, at the same time, preventing the occurrence of multistability. The method is numerically validated on the problem of tailoring the bifurcation diagram of an exothermic chemical reactor.

In this paper we consider spun dynamical systems and show that we can obtain a system on S³ × S¹ which contains any finite set of knotted and linked surfaces which are invariant surfaces for the flow.

The dynamics of map F_α,β(z) = 1/(α + βe^-z) are explored for portions of the real parameter plane where no fixed points are present on the real line. Careful tracking of the prepoles of order k and their relationship to asymptotic values yields regions in the parameter plane where attracting or elliptic cycles of period k + 2 are found. When α is fixed, it is shown for k = 0 that except for at most finitely many values of β, the 2-cycles found are indeed attracting. Numerical observations indicate that higher order cycles are also attracting, and the Julia set for two different such cases is depicted.

This paper reports the finding of an unusual three-dimensional autonomous quadratic Lorenz-like chaotic system which, surprisingly, has two stable node-type of foci as its only equilibria. The new system contains the diffusionless Lorenz system and the Burke–Shaw system, and some others, as special cases. The algebraic form of the new chaotic system is similar to the other Lorenz-type systems, but they are topologically nonequivalent. To further analyze the new system, some dynamical behaviors such as Hopf bifurcation and singularly degenerate heteroclinic and homoclinic orbits, are rigorously proved with simulation verification. Moreover, it is proved that the new system with some specified parameter values has Silnikov-type homoclinic and heteroclinic chaos.

This paper presents a Ying–Yang theory for nonlinear discrete dynamical systems considering both positive and negative iterations of discrete iterative maps. In the existing analysis, the solutions relative to "Yang" in nonlinear dynamical systems are extensively investigated. However, the solutions pertaining to "Ying" in nonlinear dynamical systems are investigated. A set of concepts on "Ying" and "Yang" in discrete dynamical systems are introduced to help one understand the hidden dynamics in nonlinear discrete dynamical systems. Based on the Ying–Yang theory, the periodic and chaotic solutions in nonlinear discrete dynamical system are discussed, and all possible, stable and unstable periodic solutions can be analytically predicted. A discrete dynamical system with the Henon map is investigated, as an example.

In this paper we discuss the possible use of chaotic signals for testing Analog-to-Digital Converters (ADCs), with particular reference to the well-known Code Density Test (CDT, also called Histogram Test). In detail, we discuss the implementation of a chaos-based discrete-time noise generator circuit, providing the theoretical analysis of its statistical characterization. The implementation of the chaos-based device is discussed with reference to a generic hardware architecture, taking into account the nonidealities introduced by the presence of noise and the variability of the circuit parameters. Based on this device, we propose a method for generating noisy samples that are distributed, over a target subinterval of the circuit output range, according to a probability density function (pdf) that can be made arbitrarily close to the ideal uniform pdf, in exchange for an acceptable reduction of the uniform-distributed samples generation rate. Theoretical results, also supported by two experiments, confirm the reliability of the proposed solution, showing that chaotic systems can represent an alternative with respect to traditional methods for the generation of signals to be used in the Code Density Test of ADCs.

A ring of discrete-time delayed neural network with self-feedback and a valid nonmonotonic activation function is explored. Coexistence of multiple stable equilibria and chaotic dynamics are demonstrated in this discrete dynamical system. Specifically, 2^m stable stationary solutions and their basins of attraction are found for a loop with m-neurons network. The theory is established by formulating parameter conditions according to a geometric observation. The networks are further confirmed to exhibit chaotic dynamics when the magnitudes of inhibitory self-feedback weights are large enough. The scenario is based on building a snapback repeller and Marotto's theorem.

In this paper, we show sufficient conditions for the existence of limit cycles in the general continuous time two-neuron autonomous CNN. We find that different types of limit cycles correspond to different regions in the template parameter space. Actually, we are able to predict the CNN behavior from the template values for the full parameter range, except for two small bounded regions.

We consider the dynamics of the three-dimensional model of the die which can bounce with dissipation on the table. It is shown that for the realistic values of the initial energy the probabilities of the die landing on the face which is the lowest one at the beginning is larger than the probabilities of landing on any other face.

We study analytically and numerically the reinjection probability density for type-II intermittency. We find a new one-parameter class of reinjection probability density where the classical uniform reinjection is a particular case. We derive a new duration probability density of the laminar phase. New characteristic relations e^β(-1 < β < 0) appear where the exponet β deepens on the reinjection probability distributions. Analytical results are in agreement with the numerical simulations.

A crisis is investigated in high dimensional chaotic systems by means of generalized cell mapping digraph (GCMD) method. The crisis happens when a hyperchaotic attractor collides with a chaotic saddle in its fractal boundary, and is called a hyperchaotic boundary crisis. In such a case, the hyperchaotic attractor together with its basin of attraction is suddenly destroyed as a control parameter passes through a critical value, leaving behind a hyperchaotic saddle in the place of the original hyperchaotic attractor in phase space after the crisis, namely, the hyperchaotic attractor is converted into an incremental portion of the hyperchaotic saddle after the collision. This hyperchaotic saddle is an invariant and nonattracting hyperchaotic set. In the hyperchaotic boundary crisis, the chaotic saddle in the boundary has a complicated pattern and plays an extremely important role. We also investigate the formation and evolution of the chaotic saddle in the fractal boundary, particularly concentrating on its discontinuous bifurcations (metamorphoses). We demonstrate that the saddle in the boundary undergoes an abrupt enlargement in its size by a collision between two saddles in basin interior and boundary. Two examples of such a hyperchaotic crisis are given in Kawakami map.

A four-dimensional continuous-time autonomous hyperchaotic system is proposed in this letter. This system is constructed by incorporating a nonlinear control to a three-dimensional continuous-time autonomous chaotic system. The hyperchaotic system is analyzed by studying the spectrum of Lyapunov exponents and the corresponding bifurcation diagram. The system exhibits chaotic, periodic, hyperchaotic behaviors for different values of a selected control parameter. Also, a simple electronic circuit is designed and implemented. Simulations and experimental observations verify the analytical results.

The dynamics of fractional-order systems have attracted increasing attention in recent years. In this paper, we numerically study the bifurcations and chaotic behaviors in the fractional-order simplified Lorenz system using the time-domain scheme. Chaos does exist in this system for a wide range of fractional orders, both less than and greater than three. Complex dynamics with interesting characteristics are presented by means of phase portraits, bifurcation diagrams and the largest Lyapunov exponent. Both the system parameter and the fractional order can be taken as bifurcation parameters, and the range of existing chaos is different for different parameters. The lowest order we found for this system to yield chaos is 2.62.

We complete the description of bifurcations for the deformation X₀ + ε₁x(x - x_*)∂_x + ε₂xz∂_x of the Lotka–Volterra vector field X₀ = x(y + 2z +15u/2)∂_x + y(x - 2z - 7u/2)∂_y - z(x + y + 4u)∂_z, u = x + y + z - 1, which was initiated in [Gołaszewski et al., 2008]. In particular, we prove existence and uniqueness of an invariant torus which appears in this family.

In this paper, we study the two-body problem that describes the motion of two-point masses in an anisotropic space under the influence of a Newtonian force-law with two relativistic correction terms. We will show that the set of initial conditions leading to collisions and ejections and leading to escapes and captures have positive measure. Using the infinity manifold, we study capture and escape solutions in the zero-energy case. We also show that the flow on the zero energy manifold of a two-body problem given by the sum of the Newtonian potential and three anisotropic perturbations (corresponding to three relativistic correction terms) is chaotic.

A discrete dynamic model accounting for both combustion and vaporization processes is proposed. In terms of different bifurcation parameters relevant to either combustion or evaporation, various bifurcation diagrams are presented. Furthermore, the corresponding Lyapunov exponent is calculated and employed to analyze the stability of the particular dynamic system. The study indicates conclusively that the evaporation process has a significant impact on the intensity and nonlinear behavior of the system of interest, vis-à-vis a model accounting for only the gaseous combustion process. Moreover, a minimum entropy control method is employed to control the chaotic behavior inherent to the system of interest. This algorithm is intended to be implemented for control of combustion instability numerically and experimentally to provide a basis for some of the control methodologies employed in the literature.

In this paper, we consider double zero singularity of a symmetric BAM neural network model with a time delay. Based on the normal form approach and the center manifold theory, we obtain the normal form on the centre manifold with double zero singularity. Some numerical simulations support our analysis results.

One of the theoretical approaches to study spatially-extended ecosystems is given by metapopulation models, which consider fragmented populations inhabiting discrete patches linked by migration. Most of the metapopulation models assume exponential growth of the local populations and few works have explored the role of cooperation in fragmented ecosystems. In this letter, we study the dynamics and the bifurcation scenarios of a minimal, two-patch metapopulation Turing-like model given by nonlinear differential equations with an autocatalytic reaction term together with diffusion. We also analyze the extinction transients of the metapopulations focusing on the effect of coupling two local populations undergoing delayed transition phenomena due to ghost saddle remnants. We find that increasing diffusion rates enhance the delaying capacity of the ghosts. We finally propose the saddle remnant as a new class of transient generator mechanism for ecological systems.

During quiet standing, the human body continuously moves about an upright posture in an erratic fashion. Many researchers characterize postural fluctuations as a stochastic process while some others suggest chaotic dynamics for postural sway. In this study, first we examined these assumptions using principles of chaos theory in normal healthy and in patients with deteriorated postural control mechanisms. Next, we compared the ability of a nonlinear dynamics quantifier correlation dimension to that of a linear measure standard deviation to describe variability of healthy and deteriorated postural control mechanisms during quiet standing. Our findings did not provide convincing evidence for existence of low dimensional chaos within normal and abnormal sway dynamics but support the notion that postural fluctuations time series are distinguishable from these generated by a random process. The results indicated that although linear variability measures discriminated well between groups, they did not provide any information about the structure of postural fluctuations. Calculated correlation dimension as a complexity measure which describes spatio temporal organization of time series may be useful in this regard.

In this paper, we numerically show that the dynamics of the HIV system is sensitive to both the initial condition and the system parameters. These phenomena imply that the system is chaotic and exhibits a bifurcation behavior. To control the system, we propose to initiate an HIV therapy based on both the concentration of the HIV-1 viral load and the ratio of the CD4 lymphocyte population to the CD8 lymphocyte population. If the concentration of the HIV-1 viral load is higher than a threshold, then the first type of therapy will be applied. If the concentration of the HIV-1 viral load is lower than or equal to the threshold and the ratio of the CD4 lymphocyte population to the CD8 lymphocyte population is greater than another threshold, then the second type of therapy will be applied. Otherwise, no therapy will be applied. The advantages of the proposed control strategy are that the therapy can be stopped under certain conditions, while the state variables of the overall system is asymptotically stable with fast convergent rate, the concentration of the controlled HIV-1 viral load is monotonic decreasing, as well as the positivity constraint of the system states and that of the dose concentration is guaranteed to be satisfied. Computer numerical simulation results are presented for an illustration.