International Journal of Bifurcation and Chaos

The standard spatial formulation of the Prisoner's Dilemma is a historic (memoryless, i.e. only results generated in the last round are taken into account in deciding the next choice), and binary (players are confined to fully cooperating or defecting). The spatial historic model admitting intermediate degrees of cooperation (fuzzy model) is studied in this work. Comparisons are made between the binary and fuzzy models. The effect of errors and discounting is also assessed.

This investigation will describe the spatial disorder of one-dimensional Cellular Neural Networks (CNN). The steady state solutions of the one-dimensional CNN can be replaced as an iteration map which is one dimensional under certain parameters. Then, the maps are chaotic and the spatial entropy of the steady state solutions is a three-dimensional devil-staircase like function.

We provide an elementary direct computation of the classical criterion governing the (linear) phase stability under inhomogeneous perturbations of a spatially homogeneous periodic orbit close to a spatially homogeneous supercritical Hopf bifurcation.

A harmonic oscillator with two discrete time delays is considered. The local stability of the zero solution of this equation is investigated by analyzing the corresponding transcendental characteristic equation of its linearized equation and employing the Nyquist criterion. Some general stability criteria involving the delays and the system parameters are derived. By choosing one of the delays as a bifurcation parameter, the model is found to undergo a sequence of Hopf bifurcation. The direction and stability of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Resonant codimension-two bifurcation is also found to occur in this model. A complete description is given to the location of points in the parameter space at which the transcendental characteristic equation possesses two pairs of pure imaginary roots, ±iω₁, ±iω₂ with ω₁:ω₂ = m:n, where m and n are positive integers. Some numerical examples are finally given for justifying the theoretical results.

In this paper we study the qualitative behavior of a predator–prey system with nonmonotonic functional response. The system undergoes a series of bifurcations including the saddle-node bifurcation, the supercritical Hopf bifurcation, and the homoclinic bifurcation. For different parameter values the system could have a limit cycle or a homoclinic loop, or exhibit the so-called "paradox of enrichment" phenomenon. In the generic case, the model has the bifurcation of cusp-type codimension two (i.e. the Bogdanov–Takens bifurcation) but no bifurcations of codimension three.

We show how coupling of local optimization processes can lead to better solutions than multistart local optimization consisting of independent runs. This is achieved by minimizing the average energy cost of the ensemble, subject to synchronization constraints between the state vectors of the individual local minimizers. From an augmented Lagrangian which incorporates the synchronization constraints both as soft and hard constraints, a network is derived wherein the local minimizers interact and exchange information through the synchronization constraints. From the viewpoint of neural networks, the array can be considered as a Lagrange programming network for continuous optimization and as a cellular neural network (CNN). The penalty weights associated with the soft state synchronization constraints follow from the solution to a linear program. This shows that the energy cost of the ensemble should maximally decrease. In this way successful local minimizers can implicitly impose their state to the others through a mechanism of master–slave dynamics resulting in a cooperative search mechanism. Improved information spreading within the ensemble is obtained by applying the concept of small-world networks. We illustrate the new optimization method on two different problems: supervized learning of multilayer perceptrons and optimization of Lennard–Jones clusters. The initial distribution of the local minimizers plays an important role. For the training of multilayer perceptrons this is related to the choice of the prior on the interconnection weights in Bayesian learning methods. Depending on the choice of this initial distribution, coupled local minimizers (CLM) can avoid overfitting and produce good generalization, i.e. reach a state of intelligence. In potential energy surface optimization of Lennard–Jones clusters, this choice is equally important. In this case it can be related to considering a confining potential. This work suggests, in an interdisciplinary context, the importance of information exchange and state synchronization within ensembles, towards issues such as evolution, collective behavior, optimality and intelligence.

Controlling chaos has been an extremely active area of research in applied dynamical systems, following the introduction of the Ott, Grebogi, Yorke (OGY) technique in 1990 [Ott et al., 1990], but most of this research based on parametric feedback control uses local techniques. Associated with a dynamical system which pushes forward initial conditions in time, transfer operators, including the Frobenius–Perron operator, are associated dynamical systems which push forward ensemble distributions of initial conditions. In [Bollt, 2000a, 2000b; Bollt & Kostelich, 1998], we have shown that such global representations of a discrete dynamical system are useful in controlling certain aspects of a chaotic dynamical system which could only be accessible through such a global representation. Such aspects include invariant measure targeting, as well as orbit targeting. In this paper, we develop techniques to show that our previously discrete time techniques are accessible also to a differential equation. We focus on the Duffing oscillator as an example. We also show that a recent extension of our techniques by Góra and Boyarsky [1999] can be further simplified and represented in a convenient and compact way by using a tensor product.

In this paper, bifurcation analysis of the dynamic response of active magnetic bearing (AMB) with flexible rotor is presented, which includes the coupling effect between X-Y directions caused by rotational motion. The AMB systems include many nonlinear factors, such as mass imbalance, mass ratio with disk and journal, material property of shaft, and magnetic forces, etc., and its dynamic behavior are inherently nonlinear. Bifurcation diagrams of shaft journal's center, trajectories, power spectra, and Poincaré maps are used to analyze the dynamic behavior of the AMB system under different operational conditions. The key factors affecting the dynamic characteristics of the AMB system are identified. It will be beneficial to the design of AMB system.

A two-degree-of-freedom model of a nonlinear vibration absorber is considered in this paper. Both the global bifurcations and chaotic dynamics of the nonlinear vibration absorber are investigated. The nonlinear equations of motion of this model are derived. The method of multiple scales is used to find the averaged equations. Based on the averaged equations, the theory of normal form is used to obtain the explicit expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues by Maple software program. The fast and slow modes may simultaneously exist in the averaged equations. On the basis of the normal form, the global bifurcation and the chaotic dynamics of the nonlinear vibration absorber are analyzed by a global perturbation method developed by Kovacic and Wiggins. The chaotic motion of this model is also found by numerical simulation.

The work deals with a switching mode control of a magnetic levitation system that possesses a symmetrical geometry similar to that usually found in magnetic bearings. The control depends on switching the current in two attracting electromagnets installed on both sides of the suspended object. The underlying idea is to use merely discrete (noncontinuous) position sensors for the control. It is unfortunate that relying merely on one such sensor cannot enable stabilization. We observe, however, that two such sensors (mounted symmetrically on both sides of the symmetry plane) are sufficient for stabilizing the system. The dynamic features of the system are considered theoretically and also investigated by simulation work. An interesting dynamics that is associated with chaotic behavior is revealed. Namely, the region of stability of the suspended object is characterized by being related to a strange attractor. It is shown that the chaotic dynamic behavior is closely similar to that exhibited by certain purely electronic oscillators where feedback is channelled through hysteresis. The main tool employed for assessing the present chaotic behavior is a discrete map constructed for the sequence of velocities exhibited at the switching instances.

This paper studies closures of fractal sets observed in nonlinear dynamical systems excited stochastically by switched inputs. The Duffing oscillator and the forced dumped pendulum are analyzed as examples. The dynamics of the system is characterized by a fractal set in the phase space. We can numerically construct a closure that encloses the fractal set. Furthermore, it is shown that the closure is a limit cycle attractor of a dynamical system defined by the switching manifold.

A method is presented for detecting weak coupling between (chaotic) dynamical systems below the threshold of (generalized) synchronization. This approach is based on reconstruction of mixed states consisting of delayed samples taken from simultaneously measured time series of both systems.

A nonlinear communication scheme is presented. It is shown that a family of equations can be used for encoding and that synchronization can be maintained even though parameters are varied at the encoding end. It is also demonstrated that the idea of using the sensitive dependence on initial conditions of chaotic system for communication purposes can also be implemented for nonchaotic systems near the common boundary of basins of attraction for multiple attractors. Numerical simulations are also included.

In this letter, the impulsive feedback synchronization method is suggested. Synchronization condition is given by investigating the mechanism of the impulsive feedback synchronization. Furthermore, we consider the influences of noise and parameter mismatch since they inevitably exist in the practical experiment. Finally, a visual circuit implementation of impulsive feedback synchronization is performed. The research demonstrates that impulsive operator can spread frequency spectrum of chaotic solution. Impulsive feedback can synchronize chaotic systems and has the robustness to noise and parameter mismatch. It shows the potential application in chaotic communication.

Applying invariant manifold theorem, we study the existence of generalized synchronization of a coupled system, with local systems being different sine circle maps. We specify a range of parameters for which the coupled system achieves generalized synchronization. We also investigate the relation between generalized synchronization, predictability and equivalence of dynamical systems. If the parameters are restricted in the specified range, then all the subsystems are topologically equivalent, and each subsystem is predictable from any other subsystem. Moreover, these subsystems are frequency locked even if the frequencies are greatly different in the absence of coupling. If the local systems are identical without coupling, then the widths of the phase-locked intervals of the coupled system are the same as those of the individual map and are independent of the coupling strength.

The synchronization of spatiotemporal chaotic system can be achieved by a novel approach — pinning and periodic impulsive method. Two kinds of pinning, regular or random, are considered. Our numerical experiments show that the relationship between the synchronization time and the pinning interval is approximately square power, and in some range of parameters the coefficient of power functional is proportional to the impulsive interval. The convergence to the synchronized state is faster in the case of nonidentical array of coupled pendula than that of identical array of coupled pendula.

The method of direct perturbation is applied to a rf-driven Josephson junction with strong and quadratic damping resistor. Perturbed correction to the heteroclinic orbit is constructed and its boundedness conditions are established to contain the Melnikov criterion for the onset of chaos. The result shows that the corrected heteroclinic orbit is unbounded, unless it is chaotic. The analytically deterministic chaotic solution exposes the incomputability of chaotic orbits, which is numerically demonstrated.

This paper experimentally verifies that one-way three-coupled chaotic maps implemented by electronic circuits can be stabilized by a decentralized delayed-feedback control. The main part of each map site is a chaos IC chip (NJH1101) which includes a piecewise nonlinear map, delay devices and clock generator. The delayed-feedback controller is also implemented by electronic devices. Our control experiments succeed in stabilizing a fixed point of the coupled maps.

The Continuation of Invariant Subspaces (CIS) algorithm [Demmel et al., 2001; Dieci & Friedman, 2001] produces a smooth orthogonal similarity transformation to block triangular form of a parameter dependent matrix A(s). The CIS algorithm is an adaptation of an iterative refinement technique for improving the accuracy of computed invariant subspaces.

On the basis of the CIS algorithm, new test functions are defined for an improved, more reliable detection of bifurcations in problems with simple bifurcations near multiple bifurcations, problems with multiple bifurcations, and problems with symmetries. Illustrative examples are given.

A concrete numerical example of Z₃-equivariant planar perturbed Hamiltonian vector field of fifth degree having 23 limit cycles and a configuration of compound eyes are given, by using the bifurcation theory of planar dynamical systems and the method of detection functions. It gives rise to the conclusion: the Hilbert number H(5) ≥ 23 for the second part of Hilbert's 16th problem.

In this paper we study the cubic nilpotent systems having an isolated singular point at the origin with degenerate infinity and give a characterization of those that accept a certain type of analytical first integrals. For these systems we give, also, a characterization of the singular point at the origin; in particular we see that the origin is never a center.