Small Attractor Research Articles

Experimentally Viable Techniques for Accessing Coexisting Attractors Correlated with Lyapunov Exponents

Universal, predictive attractor patterns configured by Lyapunov exponents (LEs) as a function of the control parameter are shown to characterize periodic windows in chaos just as in attractors, using a coherent model of the laser with injected signal. One such predictive pattern, the symmetric-like bubble, foretells of an imminent bifurcation. With a slight decrease in the gain parameter, we find the symmetric-like bubble changes to a curved trajectory of two equal LEs in one attractor, while an increase in the gain reverses this process in another attractor. We generalize the power-shift method for accessing coexisting attractors or periodic windows by augmenting the technique with an interim parameter shift that optimizes attractor retrieval. We choose the gain as our parameter to interim shift. When interim gain-shift results are compared with LE patterns for a specific gain, we find critical points on the LE spectra where the attractor is unlikely to survive the gain shift. Noise and lag effects obscure the power shift minimally for large domain attractors. Small domain attractors are less accessible. The power-shift method in conjunction with the interim parameter shift is attractive because it can be experimentally applied without significant or long-lasting modifications to the experimental system.

Necessary/Sufficient Conditions for the Chaos Game to Render Quasi Attractors

We explore when the chaos game renders a quasi attractor of an iterated function system (IFS); we emphasize the role of the set in which the chaos game is initialized. For IFS with lower semicontinuous Hutchinson-Barnsley operator we find that these initial points necessarily belong to the largest invariant set of the weak basin of the attractor. Under additional assumptions, namely the attractor is a stable small attractor and the space compact, we find that the probabilistic chaos game with initial points in the largest invariant set of the weak basin does in fact render the attractor. We note that this result applies to IFS which contain discontinuous functions. Under the same additional assumptions we provide the same conclusion for the disjunctive chaos game played with an evenly continuous IFS (this is the topological version of equicontinuous IFS).

Small and minimal attractors of an IFS

Lasota and Myjak demonstrated that one can study attractors to an iterated function system (IFS) possibly containing discontinuous functions. We continue this line of thought by working with IFS with a lower semicontinuous Hutchinson-Barnsley operator and two new (but not significantly different) types of attractors, the small attractor (the smallest closed nonempty invariant set) and minimal attractors (a minimal closed nonempty invariant set). We characterize exactly when an IFS possesses a small attractor and provide several practically verifiable sufficient conditions for this. To study minimal attractors we create the notion of the weak basin; we show a minimal attractor behaves much like a small attractor on its weak basin and that the weak basin is the largest set in which a minimal attractor behaves like this. We then give a characterization of when the weak basin is open. Further, we show that, when the iterates of the Hutchinson-Barnsley operator of an IFS form an equicontinuous set of multifunctions, both the weak basin and the point wise basin (of a point wise attractor) is closed.

Dynamics of a new composite four–Scroll chaotic system

A new three-dimensional autonomous chaotic system is proposed. This new system can generate single-scroll, double-scroll, three-scroll and four-scroll attractors under different system parameters. Particularly, it can generate a four-scroll chaotic attractor composed of a large Chua-like attractor and a small Lorenz-like attractor. And the system can also generate a nested three-scroll attractor and the multi-double-scroll chaotic attractor. In addition, the system possesses the chaotic state transition, and the number of scrolls will change in the state transition process. The formation mechanism of the composite four-scroll chaotic attractor is analyzed in detail. The dynamic analysis methods include time series, 0–1 test chart, phase diagram, bifurcation diagram and Lyapunov exponents are used to describe some basic dynamics behaviors of the proposed system.

Biparametric investigation of the general standard map: multistability and global bifurcations

We investigate multistability and global bifurcations in the general standard map, a biparametric two-dimensional map. Departing from the conservative case of the map, we describe the evolution of periodic solutions and their basins of attraction as dissipation builds up, paying special attention on how the biparametric variation affects multistability. We examine general and specific phenomena and behavior for three distinct dynamical regimes, namely small, moderate, and large damping and different forcing amplitudes. Also, we report numerically the mechanism of global bifurcations associated to small chaotic attractors in the multistable system. Several global bifurcations are investigated as dissipation increases. Specifically, through the characterization of an interior, a merging and a boundary crisis, we study the crucial role played by fundamental hyperbolic invariant structures, such as unstable periodic orbits and their stable and unstable invariant manifolds, in the mechanisms by which the phase space is globally transformed.

Detecting small attractors of large Boolean networks by function-reduction-based strategy.

Boolean networks (BNs) are widely used to model gene regulatory networks and to design therapeutic intervention strategies to affect the long-term behaviour of systems. A central aim of Boolean-network analysis is to find attractors that correspond to various cellular states, such as cell types or the stage of cell differentiation. This problem is NP-hard and various algorithms have been used to tackle it with considerable success. The idea is that a singleton attractor corresponds to n consistent subsequences in the truth table. To find these subsequences, the authors gradually reduce the entire truth table of Boolean functions by extending a partial gene activity profile (GAP). Not only does this process delete inconsistent subsequences in truth tables, it also directly determines values for some nodes not extended, which means it can abandon the partial GAPs that cannot lead to an attractor as early as possible. The results of simulation show that the proposed algorithm can detect small attractors with length p = 4 in BNs of up to 200 nodes with average indegree K = 2.

The isolation characteristics of an archetypal dynamical model with stable-quasi-zero-stiffness

In this paper, a single-degree-of-freedom geometrically nonlinear oscillator with stable-quasi-zero-stiffness (SQZS) is presented, which can be extensively applied in vibration isolation due to its high static load bearing capacity and low dynamic stiffness. This model comprises a lumped mass denoeing the isolated object and a pair of horizontal springs providing negative stiffness paralleled with a vertical linear spring to bear the load. The equation of motion of the system is formulated with an originally irrational nonlinearity based upon SD oscillator instead of the conventionally approximate Duffing system of polynomial type, which will produce results with a high precision unquestionably, especially for the prediction of a large displacement behaviour. The frequency response characteristics particularly for transmissibility of the model, subjected to harmonic forcing and vibrating base, are obtained by using an extended averaging approach to achieve the parameter optimization for maximum frequency band of isolation. Furthermore, numerical simulations are carried out to detect the complex dynamical phenomena of periodic, chaotic motions and coexistence of multiple solutions, and so on, in addition to verifying the analytical results. Finally, an interesting strategy is proposed to extend the frequency band of isolation by controlling the initial value within the attraction basin of a small amplitude attractor, which can be utilized for an effective isolation. The results presented herein provide an insight of dynamics into the SQZS mechanism for its application in vibration engineering.

Multistability in the Lorenz System: A Broken Butterfly

In this paper, the dynamical behavior of the Lorenz system is examined in a previously unexplored region of parameter space, in particular, where r is zero and b is negative. For certain values of the parameters, the classic butterfly attractor is broken into a symmetric pair of strange attractors, or it shrinks into a small attractor basin intermingled with the basins of a symmetric pair of limit cycles, which means that the system is bistable or tristable under certain conditions. Although the resulting system is no longer a plausible model of fluid convection, it may have application to other physical systems.

Strange Attractors in a Three-Dimensional Autonomous Polynomial Equation

We studied a three-dimensional autonomous polynomial equation. This equation is with a parameter, which we denote as μ. At μ = 0, the system has two saddles, the stable and unstable manifolds of which coincide. We present a comprehensive study on the dynamics of the system for small μ ≠ 0 in a small neighborhood of the unperturbed stable and unstable manifolds, where one of the heteroclinic connections of the two saddles are broken by small perturbations and strange attractors are created.

Detecting variation in chaotic attractors

If the output of an experiment is a chaotic signal, it may be useful to detect small changes in the signal, but there are a limited number of ways to compare signals from chaotic systems, and most known methods are not robust in the presence of noise. One may calculate dimension or Lyapunov exponents from the signal, or construct a synchronizing model, but all of these are only useful in low noise situations. I introduce a method for detecting small variations in a chaotic attractor based on directly calculating the difference between vector fields in phase space. The differences are found by comparing close strands in phase space, rather than close neighbors. The use of strands makes the method more robust to noise and more sensitive to small attractor differences.

Coexistence of invariant sets with and without SRB measures in Hénon family**Dedicated to the memory of Floris Takens (12 November 1940–20 June 2010).

Let {fa,b} be the (original) Hénon family. In this paper, we show that, for any b near 0, there exists a closed interval Jb which contains a dense subset J′ such that, for any a ∊ J′, fa,b has a quadratic homoclinic tangency associated with a saddle fixed point of fa,b which unfolds generically with respect to the one-parameter family . By applying this result, we prove that Jb contains a residual subset such that, for any , fa,b admits the Newhouse phenomenon. Moreover, the interval Jb contains a dense subset such that, for any , fa,b has a large homoclinic set without SRB measure and a small strange attractor with SRB measure simultaneously.

Excitability of periodic and chaotic attractors in semiconductor lasers with optoelectronic feedback

The dynamics of a semiconductor laser with AC-coupled nonlinear optoelectronic feedback has been experimentally studied. A period doubling sequence of small periodic and chaotic attractors is observed, each of them displaying excitable features. This scenario is found also in a simplified physical model of the system, thus extending the concept of excitability, usually associated to fixed points, also to the case of higher-dimensional attractors.

Distribution and enumeration of attractors in probabilistic Boolean networks

Many mathematical models for gene regulatory networks have been proposed. In this study, the authors study attractors in probabilistic Boolean networks (PBNs). They study the expected number of singleton attractors in a PBN and show that it is (2 - (1/2)(L-1))(n), where n is the number of nodes in a PBN and L is the number of Boolean functions assigned to each node. In the case of L=2, this number is simplified into 1.5(n). It is an interesting result because it is known that the expected number of singleton attractors in a Boolean network (BN) is 1. Then, we present algorithms for identifying singleton and small attractors and perform both theoretical and computational analyses on their average case time complexities. For example, the average case time complexities for identifying singleton attractors of a PBN with L=2 and L=3 are O(1.601(n)) and O(1.763(n)), respectively. The results of computational experiments suggest that these algorithms are much more efficient than the naive algorithm that examines all possible 2(n) states.

Chaotically Spiking Canards in an Excitable System with 2D Inertial Fast Manifolds

We introduce a new class of excitable systems with two-dimensional fast dynamics that includes inertia. A novel transition from excitability to relaxation oscillations is discovered where the usual Hopf bifurcation is followed by a cascade of period doubled and chaotic small excitable attractors and, as they grow, by a new type of canard explosion where a small chaotic background erratically but deterministically triggers excitable spikes. This scenario is also found in a model for a nonlinear Fabry-Perot cavity with one pendular mirror.

Algorithms for Finding Small Attractors in Boolean Networks

A Boolean network is a model used to study the interactions between different genes in genetic regulatory networks. In this paper, we present several algorithms using gene ordering and feedback vertex sets to identify singleton attractors and small attractors in Boolean networks. We analyze the average case time complexities of some of the proposed algorithms. For instance, it is shown that the outdegree-based ordering algorithm for finding singleton attractors works in O(1.19(n)) time for K = 2, which is much faster than the naive O(2(n)) time algorithm, where n is the number of genes and K is the maximum indegree. We performed extensive computational experiments on these algorithms, which resulted in good agreement with theoretical results. In contrast, we give a simple and complete proof for showing that finding an attractor with the shortest period is NP-hard.

一般化学習ネットワークの教師付き学習による連想記憶モデルの構築

Since the first neuron model was proposed, a lot of Neural Networks have been devised and been put into lots of practical uses. It is also true in the field of associative memory. Although so many useful memory models have been devised, there are still some problems, such as the limitation of storage capacity or too small attractor size to be stored.In this paper, to solve the above problems, a novel associative memory is proposed. Its unique features are, (1) the memory network is obtained by training network parameters, (2) the size of the attractor of each stored memory can be controlled, and (3) some redundant nodes are introduced into the memory network in order to increase the storage capacity.It is clarified from simulations that the proposed method can improve the memory functions, and can be applicable to the mutual associative memory easily.

Small scales and anisotropy in low Rm magnetohydrodynamic turbulence

In this paper, we derive estimates for size of the small scales and the attractor dimension in low Rm magnetohydrodynamic turbulence by deriving a rigorous upper bound of the dimension of the attractor representing this flow. To this end, we find an upper bound for the maximum growth rate of any n-dimensional volume of the phase space by the evolution operator associated to the Navier–Stokes equations. As explained in Constantin et al. [J. Fluid Mech. 150, 427 (1985)], the value of n for which this maximum is zero is an upper bound for the attractor dimension. In order to use this property in the more precise case of a three-dimensional periodical domain, we are led to calculate the distribution of n modes which minimizes the total (viscous and Joule) dissipation. This set of modes turns out to exhibit most of the well known properties of magnetohydrodynamic turbulence, previously obtained by heuristic considerations such as the existence of the Joule cone under strong magnetic field. The sought estimates for the small scales and attractor dimension are then obtained under no physical assumption as functions of the Hartmann and the Reynolds numbers and match the Hartmann number dependency of heuristic results. A necessary condition for the flow to be tridimensional and anisotropic (as opposed to purely two-dimensional) is also built.

On the Effect of the Machine Saturation on SSR in Power Systems

A bifurcation analysis is performed to investigate the influence of iron saturation on the complex dynamics of a heavily loaded single-machine-infinitebusbar power system modeling the characteristics of the CHOLLA# 4 generator with respect to the SOWARO station. The system has five mechanical and two electrical modes. The results show that as the compensation level increases, the operating condition loses stability via a supercritical Hopf bifurcation. As a result, the power system oscillates subsynchronously with a small limit-cycle attractor. The instability is due to the interaction of the subsynchronous electrical mode with one of the torsional modes. When the machine saturation is considered, again the system loses stability via a supercritical Hopf bifurcation; however, it occurs at a slightly lower level of compensation, indicating that the influence of machine saturation is slightly destabilizing.

A bifurcation analysis of subsynchronous oscillations in power systems

A bifurcation analysis is used to investigate the complex dynamics of a heavily loaded turbine-generator system connected to an infinite busbar through a series capacitor-compensated transmission line. It reveals the existence of self-excited subsynchronous torsional oscillations of a 5-mass rotor, which may eventually lead to the destruction of the shaft or the loss of synchronism of the generator. Specifically, we show that, as the capacitor-compensation value increases and reaches a critical value, called supercritical Hopf bifurcation, the system around the operating point undergoes small single-period oscillations with constant amplitude. This in turn results in a small limit-cycle attractor. As the compensation level increases further, the amplitude of oscillation grows until a secondary Hopf bifurcation is reached. There, the oscillations characterize themselves by two incommensurate periods and bounded amplitudes, signifying the transformation of the limit cycle into two-period quasiperiodic motion called a two-torus attractor. When the capacitor-compensation level passes a third critical value, the amplitude of oscillations becomes unbounded following the destruction of the two-torus attractor and its basin of attraction in a so-called bluesky catastrophe. Interestingly, this scenario repeats itself in the vicinity of three supercritical Hopf bifurcations. The bifurcation analysis is validated with numerical solutions of the differential equations that govern the power system.

Application of Bifurcation Theory to Subsynchronous Resonance in Power Systems

A bifurcation analysis is used to investigate the complex dynamics of a heavily loaded single-machine-infinite-busbar power system modeling the characteristics of the BOARDMAN generator with respect to the rest of the North-Western American Power System. The system has five mechanical and two electrical modes. The results show that, as the compensation level increases, the operating condition loses stability with a complex conjugate pair of eigenvalues of the Jacobian matrix crossing transversely from the left- to the right-half of the complex plane, signifying a Hopf bifurcation. As a result, the power system oscillates subsynchronously with a small limit-cycle attractor. As the compensation level increases, the limit cycle grows and then loses stability in a secondary Hopf bifurcation, resulting in the creation of a two-period quasiperiodic subsynchronous oscillation, a two-torus attractor. On further increases of the compensation level, the quasiperiodic attractor collides with its basin boundary, resulting in the destruction of the attractor and its basin boundary in a bluesky catastrophe. Consequently, there are no bounded motions. The results show that adding damper windings may induce subsynchronous resonance.