Grid multi-wing butterfly chaotic attractors generated from a new 3-D quadratic autonomous system

Due to the dynamic characteristics of the Lorenz system, multi-wing chaotic systems are still confined in the positive half-space and fail to break the threshold limit. In this paper, a new approach for generating complex grid multi-wing attractors that can break the threshold limit via a novel nonlinear modulating function is proposed from the firstly proposed double-wing chaotic system. The proposed method is different from that of classical multi-scroll chaotic attractors generated by odd-symmetric multi-segment linear functions from Chua system. The new system is autonomous and can generate various grid multi-wing butterfly chaotic attractors without requiring any external forcing, it also can produce grid multi-wing both on the xz-plane and yz-plane. Basic properties of the new system such as dissipation property, equilibrium, stability, the Lyapunov exponent spectrum and bifurcation diagram are introduced by numerical simulation, theoretical analysis and circuit experiment, which confirm that the multi-wing attractors chaotic system has more rich and complicated chaotic dynamics. Finally, a novel module-based unified circuit is designed which provides some principles and guidelines for future circuitry design and engineering application. The circuit experimental results are consistent with the numerical simulation results.&nbsp

Un attracteur étrange 3D versatile à six ailes

To create additional wings to a given strange attractor, several methods based on the heteroclinic loop or switching controls for example are applied, but complicate the approach and require the extension of the system to one or more other dimensions of the phase space. This deflects us from the objectives of research on low-dimensional chaotic systems. Remaining in this narrow area of 3D phase spaces to invent multi-wing attractors constitutes the main scope of the present paper. Indeed, we present a rapid investigation of a very simple autonomous 3D system of firts-order differential equations with a rich variety of phase portraits. This new intentionally constructed model exhibits double, four-or even six-wing strange attractors. We point out that under the influence of the scalar parameters, such versatile chaotic attractors are obtained. A similar sequence was likewise observed for the periodic behaviors. Besides, both chaotic or regular featured trajectories are found to be in bilateral agreement even when the morphology of the portrait changes. Obviously, we present the basic attributes of the system and its bifurcation diagram. Eventually, we emphasize that the study of the relationship between the written differential equations and the observed characteristics of attractors remains undervalued.En vue de susciter l"apparition d"ailes supplémentaires pour un attracteur étrange donné, plusieurs méthodes basées sur la boucle hétéroclinique ou une commande de commutation, par exemple, sont appliquées. Toutefois, elles compliquent l'approche et nécessitent l'extension du système vers une ou plusieurs autres dimensions de l'espace des phases. Cela nous détourne des objectifs même de la recherche sur les systèmes chaotiques à faible dimension. C'est ainsi que rester dans cette zone étroite d'espaces de phase 3D pour inventer des attracteurs à ailes multiples constitue le principal sujet du présent article. En effet, nous présentons une étude rapide d’un système 3D très simple et autonome d’équations différentielles du premier ordre avec une grande variété de portraits de phase. Ce nouveau modèle construit intentionnellement présente des attracteurs étranges à deux, quatre voire même six ailes. Nous soulignons que sous l’influence des seuls paramètres scalaires, de tels attracteurs chaotiques versatiles sont obtenus. Une séquence similaire a également été observée pour les comportements périodiques. En outre, les trajectoires chaotiques ou régulières sont en accord bilatéral même lorsque la morphologie du portrait de phase change. Nous présentons évidemment les attributs de base du système et son diagramme de bifurcation. Finalement, nous soulignons que l’étude de la relation entre la formulation des équations différentielles et les caractéristiques observées des attracteurs demeure marginale dans la littérature scientifique

A new type of four-wing chaotic attractors in 3-D quadratic autonomous systems.

Abstract: In this paper, several smooth canonical 3-D continuous autonomous systems are proposed in terms of the coefficients of nonlinear terms. These systems are derived from the existing 3-D four-wing smooth continuous autonomous chaotic systems. These new systems are the simplest chaotic attractor systems which can exhibit four wings. They have the basic structure of the existing 3-D four-wing systems, which means they can be extended to the existing 3-D fourwing chaotic systems by adding some linear and/or quadratic terms. Two of these systems are analyzed. Although the two systems are similar to each other in structure, they are different in dynamics. One is sensitive to the initializations and sampling time, but another is not, which is shown by comparing Lyapunov exponents, bifurcation diagrams, and Poincaré maps

Estimating model evidence using data assimilation

We review the field of data assimilation (DA) from a Bayesian perspective and show that, in addition to its by now common application to state estimation, DA may be used for model selection. An important special case of the latter is the discrimination between a factual model–which corresponds, to the best of the modeller's knowledge, to the situation in the actual world in which a sequence of events has occurred–and a counterfactual model, in which a particular forcing or process might be absent or just quantitatively different from the actual world. Three different ensemble‐DA methods are reviewed for this purpose: the ensemble Kalman filter (EnKF), the ensemble four‐dimensional variational smoother (En‐4D‐Var), and the iterative ensemble Kalman smoother (IEnKS). An original contextual formulation of model evidence (CME) is introduced. It is shown how to apply these three methods to compute CME, using the approximated time‐dependent probability distribution functions (pdfs) each of them provide in the process of state estimation. The theoretical formulae so derived are applied to two simplified nonlinear and chaotic models: (i) the Lorenz three‐variable convection model (L63), and (ii) the Lorenz 40‐variable midlatitude atmospheric dynamics model (L95). The numerical results of these three DA‐based methods and those of an integration based on importance sampling are compared. It is found that better CME estimates are obtained by using DA, and the IEnKS method appears to be best among the DA methods. Differences among the performance of the three DA‐based methods are discussed as a function of model properties. Finally, the methodology is implemented for parameter estimation and for event attribution

Review article: Dynamical systems, algebraic topology and the climate sciences

The definition of climate itself cannot be given without a proper understanding of the key ideas of long-term behavior of a system, as provided by dynamical systems theory. Hence, it is not surprising that concepts and methods of this theory have percolated into the climate sciences as early as the 1960s. The major increase in public awareness of the socio-economic threats and opportunities of climate change has led more recently to two major developments in the climate sciences: (i) the Intergovernmental Panel on Climate Change's successive Assessment Reports and (ii) an increasing understanding of the interplay between natural climate variability and anthropogenically driven climate change. Both of these developments have benefited from remarkable technological advances in computing resources, relating throughput as well as storage, and in observational capabilities, regarding both platforms and instruments. Starting with the early contributions of nonlinear dynamics to the climate sciences, we review here the more recent contributions of (a) the theory of non-autonomous and random dynamical systems to an understanding of the interplay between natural variability and anthropogenic climate change and (b) the role of algebraic topology in shedding additional light on this interplay. The review is thus a trip leading from the applications of classical bifurcation theory to multiple possible climates to the tipping points associated with transitions from one type of climatic behavior to another in the presence of time-dependent forcing, deterministic as well as stochastic.</p

Symmetry in Chaotic Systems and Circuits

Symmetry can play an important role in the field of nonlinear systems and especially in the design of nonlinear circuits that produce chaos. Therefore, this Special Issue, titled “Symmetry in Chaotic Systems and Circuits”, presents the latest scientific advances in nonlinear chaotic systems and circuits that introduce various kinds of symmetries. Applications of chaotic systems and circuits with symmetries, or with a deliberate lack of symmetry, are also presented in this Special Issue. The volume contains 14 published papers from authors around the world. This reflects the high impact of this Special Issue

Modularity, criticality, and evolvability of a developmental gene regulatory network

The existence of discrete phenotypic traits suggests that the complex regulatory processes which produce them are functionally modular. These processes are usually represented by networks. Only modular networks can be partitioned into intelligible subcircuits able to evolve relatively independently. Traditionally, functional modularity is approximated by detection of modularity in network structure. However, the correlation between structure and function is loose. Many regulatory networks exhibit modular behaviour without structural modularity. Here we partition an experimentally tractable regulatory network—the gap gene system of dipteran insects—using an alternative approach. We show that this system, although not structurally modular, is composed of dynamical modules driving different aspects of whole-network behaviour. All these subcircuits share the same regulatory structure, but differ in components and sensitivity to regulatory interactions. Some subcircuits are in a state of criticality, while others are not, which explains the observed differential evolvability of the various expression features in the system