International Journal of Bifurcation and Chaos

Homoclinic bifurcations of both equilibria and periodic orbits are argued to be critical for understanding the dynamics of the Hindmarsh–Rose model in particular, as well as of some square-wave bursting models of neurons of the Hodgkin–Huxley type. They explain very well various transitions between the tonic spiking and bursting oscillations in the model. We present the approach that allows for constructing Poincaré return mapping via the averaging technique. We show that a modified model can exhibit the blue sky bifurcation, as well as, a bistability of the coexisting tonic spiking and bursting activities. A new technique for localizing a slow motion manifold and periodic orbits on it is also presented.

In this paper, we give a review of the Inverse Frobenius–Perron problem (IFPP): how to create chaotic maps with desired invariant densities. After describing some existing methods for solving the IFPP, we present a new and simple matrix method of doing this. We show how the invariant density and the autocorrelation properties of the maps can be controlled independently. We also give some fundamental results on switching between a number of different chaotic maps and the effect this has on the overall invariant density of the system. The invariant density of the switched system can be controlled by varying the probabilities of choosing each individual map. Finally, we present an interesting application of the matrix method to image generation, by synthesizing a two-dimensional map, which when iterated, generates a well-known image.

The localization of critical parameter sets called bifurcations is often a central task of the analysis of a nonlinear dynamical system. Bifurcations of codimension 1 that can be directly observed in nature and experiments form surfaces in three-dimensional parameter spaces. In this paper, we propose an algorithm that combines adaptive triangulation with the theory of complex systems to compute and visualize such bifurcation surfaces in a very efficient way. The visualization can enhance the qualitative understanding of a system. Moreover, it can help to quickly locate more complex bifurcation situations corresponding to bifurcations of higher codimension at the intersections of bifurcation surfaces. Together with the approach of generalized models the proposed algorithm enables us to gain extensive insights in the local and global dynamics not only in one special system but in whole classes of systems. To illustrate this ability we analyze three examples from different fields of science.

The dynamics of a system defined by an endomorphism is essentially different from that of a system defined by a diffeomorphism due to interaction of invariant objects with the so-called critical locus. A planar endomorphism typically folds the phase space along curves J₀ where the Jacobian of the map is singular. The critical locus, denoted J₁, is the image of J₀. It is often only piecewise smooth due to the presence of isolated cusp points that are persistent under perturbation. We investigate what happens when the stable set W_s of a fixed point or periodic orbit interacts with J₁ near such a cusp point C₁. Our approach is in the spirit of bifurcation theory, and we classify the different unfoldings of the codimension-two singularity where the curve W_s is tangent to J₁ exactly at C₁. The analysis uses a local normal-form setup that identifies the possible local phase portraits. These local phase portraits give rise to different global manifestations of the behavior as organized by five different global bifurcation diagrams.

Bazykin proposed a Lotka–Volterra-type ecological model that accounts for simplified territoriality, which neither depends on territory size nor on food availability. In this study, we describe the global dynamics of the Bazykin model using analytical and numerical methods. We specifically focus on the effects of mutual predator interference and the prey carrying capacity since the variability of each could have especially dramatic ecological repercussions. The model displays a broad array of complex dynamics in space and time; for instance, we find the coexistence of a limit cycle and a steady state, and bistability of steady states. We also characterize super- and subcritical Poincaré–Andronov–Hopf bifurcations and a Bogdanov–Takens bifurcation. To illustrate the system's ability to naturally shift from stable to unstable dynamics, we construct bursting solutions, which depend on the slow dynamics of the carrying capacity. We also consider the stabilizing effect of the intraspecies interaction parameter, without which the system only shows either a stable steady state or oscillatory solutions with large amplitudes. We argue that this large amplitude behavior is the source of chaotic behavior reported in systems that use the MacArthur–Rosenzweig model to describe food-chain dynamics. Finally, we find the sufficient conditions in parameter space for Turing patterns and obtain the so-called "back-eye" pattern and localized structures.

Two methodologies are presented for the numerical approximation of the "domain of stability" of nonlinear conservative maps: (a) the Evolutionary Estimation of the Domain of Stability (EEDS) and (b) the Evolutionary Frequency Optimization (EFO), optimizing certain frequency parameters of these maps so that the domain of stability encompasses the maximum possible "volume" of bounded motion, known in the accelerator literature as the dynamic aperture. The central components of the proposed approaches are: The Differential Evolution algorithm (DE) based on concepts of Computational Intelligence and the method of the Smaller ALignment Index (SALI) used for the determination of chaotic dynamics. Initially, we give a brief description of the two methodologies and then demonstrate their usefulness by applying them to some well-known examples of 2D and 4D Hénon maps. The proposed methodologies can be easily applied to "volume" preserving maps which are not necessarily symplectic as well as to continuous dynamical systems (flows) and can also be generalized to treat conservative dynamical systems of any dimension.

In this paper we investigate the growth rate of the number of all possible paths in graphs with respect to their length in an exact analytical way. Apart from the typical rates of growth, i.e. exponential or polynomial, we identify conditions for a stretched exponential type of growth. This is made possible by combining two or more graphs over the same alphabet, in order to obtain a discrete dynamical system generated by a triangular map, which can also be interpreted as a discrete nonautonomous system. Since the vertices and the edges of a graph are usually used to depict the states and transitions between states of a discrete dynamical system, the combination of two (or more) graphs can be interpreted as the driving, or perturbation, of one system by another.

This paper concerns the bifurcation analysis of a recently proposed second-order impact model for forest fire prediction. The analysis is carried out with respect to four pairs of system parameters and is based on a one-dimensional piecewise-smooth map naturally induced by the two-dimensional flow. The results of the analysis confirm those reported in the literature and extend them, pointing out the different roles played by several system parameters in influencing the qualitative behavior of fire regimes.

We study the dynamics of a reaction–diffusion system composed of two mutually coupled excitable fibers. We focus on the situation in which dynamical properties of the two fibers are not identical because of the parameter difference between the fibers. Using the spatially one-dimensional FitzHugh–Nagumo equations as a model of a single excitable fiber, we show that the system exhibits a rich variety of dynamical behavior, including soliton-like collision between two pulses and recombination of a solitary pulse and synchronized pulses.

We study the family of complex maps given by F_λ(z) = zⁿ + λ/zⁿ + c where n ≥ 3 is an integer, λ is an arbitrarily small complex parameter, and c is chosen to be the center of a hyperbolic component of the corresponding Multibrot set. We focus on the structure of the Julia set for a map of this form generalizing a result of McMullen. We prove that it consists of a countable collection of Cantor sets of closed curves and an uncountable number of point components.

This paper presents an approach to control the chaotic dynamics of discrete-time (or discretizable) systems. The objective of the paper is to focus on the suppression of the chaotic dynamics and the restoration of order with a state feedback controller. The proposed control method works by targeting instantaneous measures of the Lyapunov exponents of a system and setting them to desired values. At first, the paper presents an instantaneous measure of the Lyapunov exponents; this measure is used to control the system's dynamics. Then, the formulation of the control algorithm to suppress chaos is presented. Two cases for the control structure are considered. The first case corresponds to the case when the number of control inputs is equal to the number of states; the second case corresponds to the case when the number of control inputs is less than the number of states. The Lorenz system, the smooth Chua Oscillator system, the Rössler-hyperchaos system and a fourth order chaotic oscillator system are used as examples to illustrate the proposed control scheme. The simulation results show the efficacy of the proposed control approach.

In this paper, a more general third-order chaotic system with attraction/repulsion function is introduced on the basis of [Duan et al., 2005]. A gallery of chaotic attractors, bifurcation diagrams and Lyapunov exponent spectra are presented to show the interesting phenomena of the given system. Based on the absolute stability theory and linear matrix inequality (LMI), a simple method of chaos control for the system is proposed and a stabilizing controller is derived such that chaos oscillations of the system disappear and all chaotic trajectories of it are led to certain equilibrium. Numerical simulations are provided to illustrate the efficiency of the proposed method.

Using the examples of an excitable chemical system (the Belousov–Zhabotinsky medium) and plasmodium of Physarum polycephalum we show that universal computation in a geometrically unconstrained medium is only possible when resources (excitability or concentration of nutrients) are limited. In situations of limited resources the systems studied develop traveling localizations. These localizations are the elementary units of dynamical logical circuits in collision-based computing architectures.

In this paper, chaos in the smooth Chua's equation is revisited. To confirm the chaotic behavior in the smooth Chua's equation demonstrated in numerical studies, we resort to Poincaré section and Poincaré map technique and present a computer assisted verification of existence of horseshoe chaos by virtue of topological horseshoes theory.

The active control of the unstable synchronization manifold in a shift-invariant ring of N mutually coupled chaotic oscillators is investigated. After deriving the bifurcation structures and chaotic states in the single oscillator, we find the regime of coupling parameters leading to stable and unstable synchronization phenomena in the ring, using the Master stability function approach with the transverse Lyapunov exponents. The active control technique is applied on the mutually coupled chaotic systems to suppress unstable synchronization states. We derive the range of control gain parameters which leads to a successful control and the stability of the control design. The effects of the amplitude of the parametric perturbations on the stability boundaries of the controlled unstable synchronization process are also studied.

Synchronization of chaotic systems with unknown parameters, or namely auto-synchronization, has aroused a lot of interest in the past few years. In this paper, we are interested in solving one of the critical cases in Lorenz system. Based on the past researches, it was concluded that the auto-synchronization of Lorenz system was inachievable if the unknown parameters resided in all of its three dynamical equations while only the first state was measurable. However, as demonstrated in this paper, this problem can be resolved by an adaptive observer following simple design rules. The design is justified with the conditional Lyapunov exponents, and its effectiveness is clearly illustrated in simulations.

This study addresses the adaptive synchronization of a class of uncertain chaotic systems in the drive-response framework. For a class of uncertain chaotic systems with parameter mismatch and external disturbances, a robust adaptive observer based on the response system is constructed to practically synchronize the uncertain drive chaotic system. Lyapunov stability theory ensures the practical synchronization between the drive and response systems even if Lipschitz constants on function matrices and bounds on uncertainties are unknown. Numerical simulation of two illustrative examples are given to verify the effectiveness of the proposed method.

Switched buffer systems are discretized switched flow systems. They are discrete, but can be handled with the same mathematical framework as hybrid systems. In particular, we show that switched buffer systems form discrete strange billiards. For any initialization, discrete strange billiards lead to a finite periodic attractor. The periods and the buffer switching rates are the most important performance measures for the practical use of switched buffer systems. Both measures depend on the sum of objects in the buffers. For each object sum multiple suboptimal attractors may exist. Onto which attractor the system converges depends on the initial buffer levels. We propose a switched Kanban control algorithm that forces the system onto the attractor with minimum buffer switching rate independently of the initialization of the buffer levels. This algorithm needs to perform only one single additional buffer switching operation. Also Heijunka control of switched buffer systems is considered in the framework of strange billiards. We show that Heijunka forms equilateral triangles in the state space. Heijunka is able to yield low buffer switching rates, but only if the buffer levels are optimally initialized. For suboptimal buffer level initializations very high (bad) buffer switching rates can be achieved. In an illustrative example, our proposed switched Kanban control algorithm outperforms conventional Kanban control by about 29% and Heijunka control by about 5%.

This paper studies the bifurcation and chaos phenomena in average queue length in a developed Transmission Control Protocol (TCP) model with Random Early Detection (RED) mechanism. Bifurcation and chaos phenomena are nonlinear behavior in network systems that lead to degradation of the network performance. The TCP/RED model used is a model validated previously. In our study, only the average queue size is considered, and the results are based on analytical model rather than actual measurements. The instabilities in the model are studied numerically using the conventional nonlinear bifurcation analysis. Extending from this bifurcation analysis, a modified RED algorithm is derived to prevent the observed bifurcation and chaos regardless of the selected parameters. Our modification is for the simple scenario of a single RED router carrying only TCP traffic. The algorithm neither compromises the throughput nor the average queuing delay of the system.

The main result of this paper is a proof of existence of a nontrivial knot on any embedded template, that was left as an open question to prove in [Ghrist et al., 1997] without using the Bennequin's inequality [Ghrist et al., 1997]. This result in the branched two-manifold case, which we prove by a sequence of lemmas showing our simple template (or ones with twists) containing nontrivial knots is (are) contained in every template as a subtemplate, enables us to generalize it later in this paper to certain forms of three-templates in four-dimensional dynamical systems by simply using the technique of "spinning" the knots in the lower dimensional templates to obtain the spun knotted surfaces.

We present a simple lattice model consisting of a chain of coupled oscillators, where their masses are interconnected by linear springs and allowed to move along a common axis, as in a monorail. In the transverse direction each mass is also attached to two other springs, one on each side of the mass. The ends of these springs are kept at fixed positions. The nonlinearity in the model arises from the geometric constraints imposed on the motion of the masses, as well as from the configuration of the springs, where in the transverse directions the springs are either in the extended or compressed state depending on the position of the mass. Under these conditions we show that solitary waves (domain walls) are present in the system. In the long wavelength limit an analytical solution for these nonlinear waves is found. Numerical integrations of the equations of motion in the full discrete system are also performed to analyze the stability of the domain wall solution. Nonlinear supratransmission is also shown to exist in the model and a discussion of mechanism is presented.

In standard cellular automata the new state of a cell depends on the neighborhood configuration solely at the preceding time step. This study analyzes the effect of implementing memory in cells of cellular automata on networks with scale-free topology. As a rule, embedding memory in cells induces a notable depletion in the cell change rate, but it is not so effective in controlling the spread of damage.