Design and FPGA implementation of a memristor-based multi-scroll hyperchaotic system

In this paper, a novel memristor-based multi-scroll hyperchaotic system is proposed. Based on a voltage-controlled memristor and a modulating sine nonlinear function, a novel method is proposed to generate the multi-scroll hyperchaotic attractors. First, a multi-scroll chaotic system is constructed from a three-dimensional chaotic system by designing a modulating sine nonlinear function. Then, a voltage-controlled memristor is introduced into the above-designed multi-scroll chaotic system. Thus, a memristor-based multi-scroll hyperchaotic system is generated, and this hyperchaotic system can produce various coexisting hyperchaotic attractors with different topological structures. Moreover, different number of scrolls and different topological attractors can be obtained by varying the initial conditions of this system without changing the system parameters. The Lyapunov exponents, bifurcation diagrams and basins of attraction are given to analyze the dynamical characteristics of the multi-scroll hyperchaotic system. Besides, the FPGA-based digital implementation of the memristor-based multi-scroll hyperchaotic system is carried out. The experimental results of the FPGA-based digital circuit are displayed on the oscilloscope.

Increasing instability of a rocky intertidal meta-ecosystem

Climate change threatens to destabilize ecological communities, potentially moving them from persistently occupied “basins of attraction” to different states. Increasing variation in key ecological processes can signal impending state shifts in ecosystems. In a rocky intertidal meta-ecosystem consisting of three distinct regions spread across 260 km of the Oregon coast, we show that annually cleared sites are characterized by communities that exhibit signs of increasing destabilization (loss of resilience) over the past decade despite persistent community states. In all cases, recovery rates slowed and became more variable over time. The conditions underlying these shifts appear to be external to the system, with thermal disruptions (e.g., marine heat waves, El Niño–Southern Oscillation) and shifts in ocean currents (e.g., upwelling) being the likely proximate drivers. Although this iconic ecosystem has long appeared resistant to stress, the evidence suggests that subtle destabilization has occurred over at least the last decade.

New Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction

The purpose of this paper is to propose new modified Newton’s method for solving nonlinear equations and free from second derivative. Convergence results show that the order of convergence is four. Several numerical examples are given to illustrate that the new iterative algorithms are effective.In the end, we present the basins of attraction to observe the fractal behavior and dynamical aspects of the proposed algorithms.

Dynamical Effects of Offset Terms on a Modified Chua’s Oscillator and Its Circuit Implementation

This paper addresses the effects of offset terms on the dynamics of a modified Chua’s oscillator. The mathematical model is derived using Kirchhoff’s laws. The model is analyzed with the help of the maximal Lyapunov exponent, bifurcation diagrams, phase portraits, and basins of attraction. The investigations show that the offset terms break the symmetry of the system, generating more complex nonlinear phenomena like coexisting asymmetric bifurcations, coexisting asymmetric attractors, asymmetric double-scroll chaotic attractors and asymmetric attraction basins. Also, a hidden attractor (period-1 limit cycle) is found when varying the initial conditions. More interestingly, this latter attractor coexists with all other self-excited ones. A microcontroller-based implementation of the circuit is carried out to verify the numerical investigations.

Extended local convergence for Newton-type solver under weak conditions

We present the local convergence of a Newton-type solver for equations involving Banach space valued operators. The eighth order of convergence was shown earlier in the special case of the k-dimensional Euclidean space, using hypotheses up to the eighth derivative although these derivatives do not appear in the method. We show convergence using only the rst derivative. This way we extend the applicability of the methods. Numerical examples are used to show the convergence conditions. Finally, the basins of attraction of the method, on some test problems are presented.

Some Properties of the Basins of Attraction of the Newton’s Method for Simple Nonlinear Geodetic Systems

The research presented in this paper concerns the determination of the attraction basins of the Newton’s iterative method which was used to solve the non-linear systems of observational equations associated with the geodetic measurements. The authors considered simple observation systems corresponding to the intersections, or linear and angular resections, used in practice. The main goal was to investigate the properties of the sets of convergent in the initial points of the applied iterative method. An important issue regarding the possibility of automatic and quick selection of such points was also considered. Therefore, the answers to the questions regarding the geometric structure of the basins, their limitations, connectedness or self-similarity were sought. The research also concerned the iterative structures of the basins, i.e. maps of the number of iterations which are necessary to achieve the convergence of the Newton’s method. The determined basins were compared with the areas of convergence that result from theorems on the convergence of the Newton’s method, i.e. the conditions imposed on the eigenvalues and norms of the matrices of the studied iterative systems. One of the essential results of the research is the indication that the obtained basins of attraction contain areas resulting from the theoretical premises and their diameters can be comparable with the sizes of the analyzed geodetic structures. Consequently, in the analyzed cases it is possible to construct methods that enable quick selection of the initial starting points or automation of such selection. The paper also characterizes the global convergence mechanism of the Newton’s method for disconnected basins and, as a consequence, the non-local initial points, i.e. located far from the solution points.

A new class of two-dimensional rational maps with self-excited and hidden attractors

This paper studies a new class of two-dimensional rational maps exhibiting self-excited and hidden attractors. The mathematical model of these maps is firstly formulated by introducing a rational term. The analysis of existence and stabilities of the fixed points in these maps suggests that there are four types of fixed points, i.e., no fixed point, one single fixed point, two fixed points and a line of fixed points. To investigate the complex dynamics of these rational maps with different types of fixed points, numerical analysis tools, such as time histories, phase portraits, basins of attraction, Lyapunov exponent spectrum, Lyapunov (Kaplan-Yorke) dimension and bifurcation diagrams, are employed. Our extensive numerical simulations identify both self-excited and hidden attractors, which were rarely reported in the literature. Therefore, the multi-stability of these maps, especially the hidden one, is further explored in the present work.

A model-based strategy for quadruped running with differentiated fore- and hind leg morphologies

This article introduces a model-based strategy for a quadruped robot with differentiated fore- and hind-leg ground reaction-force patterns to generate animal-like running behavior. The proposed model comprises a rigid body and two eSLIP legs with dampers. The eccentric-SLIP (eSLIP) model extends the traditional spring-loaded inverted pendulum (SLIP) model by adding a bar to offset the spring direction. The proposed two-leg eSLIP (TL-eSLIP) model’s fore- and hind legs were designed to have the same offset magnitude but in opposite offset directions, producing different braking and thrusting force patterns. The TL-eSLIP model’s reference leg trajectories were designed based on the fixed-point motion of the eSLIP model. Additionally, the legs were clock torque-controlled to modulate leg motion and stabilize the model to follow its natural dynamics. The model’s equations for motion were derived, and the model’s dynamic behavior was simulated and analyzed. The simulation results indicate that the model with leg offsets and in either trotting or pronking has differentiated leg force patterns, and it is more stable and has larger basins of attraction than the model without leg offsets. A quadruped robot was built for experimental validation. The experimental results demonstrate that the robot with differentiated legs ran with differentiated ground reaction force patterns and ran more stably than another robot with the same leg morphology.

Plasma-wall self-organization in magnetic fusion

This paper introduces the concept of plasma-wall self-organization (PWSO) in magnetic fusion. The basic idea is the existence of a time delay in the feedback loop relating radiation and impurity production on divertor plates. Both a zero and a onedimensional description of PWSO are provided. They lead to an iterative equation whose equilibrium fixed point is unstable above some threshold. This threshold corresponds to a radiative density limit, which can be reached for a ratio of total radiated power to total input power as low as 1/2. When detachment develops and physical sputtering dominates, this limit is progressively pushed to very high values if the radiation of non-plate impurities stays low. Therefore, PWSO comes with two basins for this organization: the usual one with a density limit, and a new one with density freedom, in particular for machines using high-Z materials. Two basins of attraction of PWSO are shown to exist for the tokamak during start-up, with a high density one leading to this freedom. This basin might be reached by a proper tailoring of ECRH assisted ohmic start-up in present middle-size tokamaks, mimicking present stellarator start-up. In view of the impressive tokamak DEMO wall load challenge, it is worth considering and checking this possibility, which comes with that of more margins for ITER and of smaller reactors.