Narrow Results

This paper presents a hybrid approach based discrete Particle Swarm Optimization (PSO) and chaotic strategies for solving multi-objective task scheduling problem in cloud computing. The main purpose is to allocate the summited tasks to the available resources in the cloud environment with minimum makespan (i.e. schedule length) and processing cost while maximizing resource utilization without violating Service Level Agreement (SLA) among users and cloud providers. The main challenges faced by Particle Swarm Optimization (PSO) when used to solve scheduling problems are premature convergence and trapping into local optimum. This paper presents an enhanced Particle Swarm Optimization algorithm hybridized with Chaotic Map strategies. The proposed approach is called Enhanced Particle Swarm Optimization based Chaotic Strategies (EPSOCHO) algorithm. Our proposed approach suggests two Chaotic Map strategies: sinusoidal iterator and Lorenz attractor to enhanced PSO algorithm in order to get good convergence and diversity for optimizing the task scheduling in cloud computing. The proposed approach is simulated and implemented in Cloudsim simulator. The performance of the proposed approach is compared with the standard PSO algorithm, the improved PSO algorithm with Longest job to fastest processor (LJFP-PSO), and the improved PSO algorithm with minimum completion time (MCT-PSO) using different sizes of tasks and various benchmark datasets. The results clearly demonstrate the efficiency of the proposed approach in terms of makespan, processing cost and resources utilization.

This letter reports the finding of a new chaotic attractor in a simple three-dimensional autonomous system, which connects the Lorenz attractor and Chen's attractor and represents the transition from one to the other.

We discuss a rather new phenomenon in chaotic dynamics connected with the fact that some three-dimensional diffeomorphisms can possess wild Lorenz-type strange attractors. These attractors persist for open domains in the parameter space. In particular, we report on the existence of such domains for a three-dimensional Hénon map (a simple quadratic map with a constant Jacobian which occurs in a natural way in unfoldings of several types of homoclinic bifurcations). Among other observations, we have evidence that there are different types of Lorenz-like attractor domains in the parameter space of the 3D Hénon map. In all cases the maximal Lyapunov exponent, Λ₁, is positive. Concerning the next Lyapunov exponent, Λ₂, there are open domains where it is definitely positive, others where it is definitely negative and, finally, domains where it cannot be distinguished numerically from zero (i.e. |Λ₂| < ρ, where ρ is some tolerance ranging between 10^-5 and 10^-6). Furthermore, several other types of interesting attractors have been found in this family of 3D Hénon maps.

We investigated a choice of prototypical double-scroll attractors, namely trajectories resulting from Lorenz equations and from two sets of equations related to Chua's circuits. We found that the probability distribution for the residence times within a given scroll consists of prominent, exponentially decaying peaks. Similar peaks had been reported for stochastically resonant systems, which are driven periodically; however, in view of our results in autonomous systems, the appearance of such peaks has a higher degree of generality. In the systems we investigated, each peak corresponds to a subset of the attractor having a particular number of loops. Moreover, the sets of initial conditions leading to different peaks are interleaved in an analogous way as riddled or intermingled basins of attraction.

The Chen system of equations exhibits Lorenz, Transition, Chen and Transverse 8 type of chaotic attractors depending on the system parameters. Some authors have proposed a generalized competitive mode (GCM) technique to explain the topological difference between the Lorenz attractor and the Chen attractor. In this paper, we show a range of parameter values for which the nature of the topological attractor for the Chen system is not in accordance with that expected from GCM analysis. Instead, we find that return maps can be used to characterize the transition between different types of attractors more reliably.

A new computational technique based on the symbolic description utilizing kneading invariants is proposed, and verified for explorations of dynamical and parametric chaos in a few exemplary systems with the Lorenz attractor. The technique allows for uncovering the stunning complexity and universality of bi-parametric structures and detects their organizing centers — codimension-two T-points and separating saddles in the kneading-based scans of the iconic Lorenz equation from hydrodynamics, a normal model from mathematics, and a laser model from nonlinear optics.

We explore the multifractal, self-similar organization of heteroclinic and homoclinic bifurcations of saddle singularities in the parameter space of the Shimizu–Morioka model that exhibits the Lorenz chaotic attractor. We show that complex transformations that underlie the transitions from the Lorenz attractor to wildly chaotic dynamics are intensified by Shilnikov saddle-foci. These transformations are due to the emergence of Shilnikov flames originating from inclination-switch homoclinic bifurcations of codimension-two. We demonstrate how the original computational technique, based on the symbolic description and kneading invariants, can disclose the complexity and universality of parametric structures and their link with nonlocal bifurcations in this representative model.

It is shown that the general class of three-dimensional first-order ordinary differential equations with quadratic nonlinearities can be physically interpreted as the dynamics of a charged particle in an electromagnetic field, with a constant gradient B-field. The general class of equations is derived within the Lagrangian formalism of classical mechanics. As an application of this interpretation a new way of experimentally realizing the Lorenz chaotic attractors is proposed. The actual construction of such systems could be facilitated by existing magnetic resonance imaging technology, which already makes use of constant gradient fields, and may find applications in areas such as nuclear medicine and magnetic confinement fusion devices.

During the last decade, fractal processes and chaotic systems were widely studied in many areas of research. Chaotic systems are highly dependent on initial conditions. Small changes in initial conditions can generate widely diverging or converging outcomes for both bifurcation or attraction in chaotic systems. In this work, we present a new method on how to generate a new family of chaotic attractors by combining these with a network of fractal processes. The proposed approach in this article is based upon the construction of a new system of fractal processes.

The paper is devoted to topical issues of modern mathematical theory of dynamical chaos and its applications. At present, it is customary to assume that dynamical chaos in finite-dimensional smooth systems can exist in three different forms. This is dissipative chaos, the mathematical image of which is a strange attractor; conservative chaos, for which the entire phase space is a large “chaotic sea” with randomly spaced elliptical islands inside it; and mixed dynamics, characterized by the principal inseparability in the phase space of attractors, repellers and conservative elements of dynamics. In the present paper (which opens a series of three of our papers), elements of the theory of pseudohyperbolic attractors of multidimensional maps and flows are presented. Such attractors, as well as hyperbolic ones, are genuine strange attractors, but they allow the existence of homoclinic tangencies. We describe two principal phenomenological scenarios for the appearance of pseudohyperbolic attractors in one-parameter families of three-dimensional diffeomorphisms, and also consider some basic examples of concrete systems in which these scenarios occur. We propagandize new methods for studying pseudohyperbolic attractors (in particular, the method of saddle charts, the modified method of Lyapunov diagrams and the so-called LMP-method for verification of pseudohyperbolicity of attractors) and test them on the above examples. We show that Lorenz-like attractors in three-dimensional generalized Hénon maps and in a nonholonomic model of Celtic stone as well as figure-eight attractors in the model of Chaplygin top are genuine (pseudohyperbolic) ones. Besides, we show an example of four-dimensional Lorenz model with a wild spiral attractor of Shilnikov–Turaev type that was found recently in [Gonchenko et al., 2018].

In this article using an analytical method called Fishing principle we obtain the region of parameters, where the existence of a homoclinic orbit to a zero saddle equilibrium in the Lorenz-like system is proved. For a qualitative description of the different types of homoclinic bifurcations, a numerical analysis of the obtained region of parameters is organized, which leads to the discovery of new bifurcation scenarios.

Even if a system has only one stable equilibrium point, its dynamics can still be complicated. In the case of the equilibrium point with a complicated boundary of the attraction basin, it is probably difficult to predict the long-term behavior of the trajectory starting from a point outside but near the boundary. The exploration of such systems is helpful for deepening our understanding of the dynamics of complex systems. This paper studies a class of three-dimensional polynomial systems with a nonelementary singularity. With parameters satisfying some conditions, the asymptotic stability of the origin is proved and the complexity of the attraction basin is investigated. It is demonstrated that the boundary of the attraction basin of the origin has a fractal structure in the following sense: An invariant set homeomorphic to the well-known Lorenz strange attractor is contained in the basin boundary. Based on the non-negative function we constructed, how fast a trajectory of the system tends to the asymptotically stable nonelementary singularity is measured.

This chapter is an introduction to complexity theory (encompassing chaos — a subset of complexity), a nascent domain, although, it possesses a historical root. Some fundamental properties of chaos/complexity (including complexity mindset, nonlinearity, interconnectedness, interdependency, far-from-equilibrium, butterfly effect, determinism/in-determinism, unpredictability, bifurcation, deterministic chaotic dynamic, complex dynamic, complex adaptive dynamic, dissipation, basin of attraction, attractor, chaotic attractor, strange attractor, phase space, rugged landscape, red queen race, holism, self-organization, self-transcending constructions, scale invariance, historical dependency, constructionist hypothesis and emergence), and its development are briefly examined. In particular, the similarities (sensitive dependence on initial conditions, unpredictability) and differences between deterministic chaotic systems (DCS) and complex adaptive systems (CAS) are analyzed. The edge of emergence (2^nd critical value, a new concept) is also conceived to provide a more comprehensive explanation of the complex adaptive dynamic (CAD) and emergence. Subsequently, a simplified system spectrum is introduced to illustrate the attributes, and summarize the relationships of the various categories of common systems.

Next, the recognition that human organizations are nonlinear living systems (high finite dimensionality CAS) with adaptive and thinking agents is examined. This new comprehension indicates that a re-calibration in thinking is essential. In the human world, high levels of human intelligence/consciousness (the latent impetus that is fundamentally stability-centric) drives a redefined human adaptive and evolution dynamic encompassing better potentials of self-organization or self-transcending constructions, autocatalysis, circular causation, localized spaces/networks, hysteresis, futuristic, and emergent of new order (involving a multi-layer structure and dynamic) — vividly indicating that intelligence/consciousness-centric is extremely vital. Simultaneously, complexity associated properties/characteristics in human organizations must be better scrutinized and exploited — that is, establishing appropriate complexity-intelligence linkages is a significant necessity. In this respect, nurturing of the intelligence mindset and developing the associated paradigmatic shift is inevitable.

A distinct attempt (the basic strategic approach) of the new intelligence mindset is to organize around human intrinsic intelligence — intense intelligence-intelligence linkages that exploits human intelligence/consciousness sources individually and collectively by focusing on intelligence/consciousness-centricity, complexity-centricity, network-centricity, complexity-intelligence linkages, collective intelligence, org-consciousness, complex networks, spaces of complexity (better risk management <=> new opportunities <=> higher sustainability) and prepares for punctuation points (better crisis management <=> collectively more intelligent <=> higher resilience/sustainability) concurrently — illustrating the significance of self-organizing capability and emergence-intelligence capacity. The conceptual development introduced will serve as the basic foundation of the intelligent organization (IO) theory.