Attractor Research Articles

Periodic and chaotic behaviors of a compound pendulum driven by a horizontal periodic external force

Abstract Based on the oscillation experiments of a compound pendulum driven by a horizontal periodic external force, a nonlinear dynamical model is established and studied numerically. The periodic, quasiperiodic and chaotic orbits are found in the motions of the compound pendulum in response to several driving forces. The numerical results are in qualitative agreement with those in the experiments. It is found that
the chaotic attractor in the three-dimensional phase space displays a two-dimensional torus structure.
On a Poincare section, the chaotic attractor consists of the relative rotating core and its trailing tails.
The fractal dimension of the chaotic attractor on the Poincare section is determined by using the box-counting method. The bifurcation diagram shows that the transition of the system from periodic motion to chaos is realized by the period-doubling bifurcation. The first Feigenbaum universal constant is approximately determined in the period-doubling bifurcation process.
The competition between the inherent vibration and the external driving vibration of the system
is thus thought as the physical mechanism for leading to the complex phenomena such as the period-doubling bifurcation and chaos.
The numerical results combining with the experimental ones can provide an intuitive and in-depth understanding of chaotic phenomena, which is of great significance for the optimization design and the stability control in the engineering technology.

Basin of attraction organization in infinite-dimensional delayed systems: A stochastic basin entropy approach.

The Mackey-Glass system is a paradigmatic example of a delayed model whose dynamics is particularly complex due to, among other factors, its multistability involving the coexistence of many periodic and chaotic attractors. The prediction of the long-term dynamics is especially challenging in these systems, where the dimensionality is infinite and initial conditions must be specified as a function in a finite time interval. In this paper, we extend the recently proposed basin entropy to randomly sample arbitrarily high-dimensional spaces. By complementing this stochastic approach with the basin fraction of the attractors in the initial conditions space, we can understand the structure of the basins of attraction and how they are intermixed. The results reported here allow us to quantify the predictability giving us an idea about the long-term evolution of trajectories as a function of the initial conditions. The tools employed can result very useful in the study of complex systems of infinite dimension.

Finding critical exponents and parameter space for a family of dissipative two-dimensional mappings.

A family of dissipative two-dimensional nonlinear mappings is considered. The mapping is described by the angle and action variables and parameterized by ε controlling nonlinearity, δ controlling the amount of dissipation, and an exponent γ is a dynamic free parameter that enables a connection with various distinct dynamic systems. The Lyapunov exponents are obtained for different values of the control parameters to characterize the chaotic attractors. We investigated the time evolution for the stationary state at period-doubling bifurcations. The convergence to the stationary state is made using a robust homogeneous and generalized function at the bifurcation parameter, which leads us to obtain a set of universal critical exponents. The parameter space of the mapping is investigated, and tangent, period-doubling, pitchfork, and cusp bifurcations are found, and a street of saddle-area and spring-area structures is observed.

Extremely multi-stable grid-scroll memristive chaotic system with omni-directional extended attractors and application of weak signal detection

The extreme multi-stability and omni-directional expansion of the attractors in chaotic systems play an indispensable role in the chaos-based engineering applications. In this study, a novel extremely multi-stable grid-scroll memristive chaotic system with omni-directional extended attractors is proposed, and applies it to weak signal detection. Firstly, a flux-controlled non-volatile memristor with a typical 8-shaped pinched hysteresis loop, and three piecewise linear functions that can promote the position offset of the attractor are designed. On this basis, we construct a novel grid-scroll memristive chaotic system with infinite equilibrium points, which can generate n × n × n-scroll attractors. The piecewise linear functions lead to the infinite expansion of the chaotic attractor in the x, y, and z directions, which also indirectly causes the position offset in the w direction. This phenomenon has not been discovered in existing researches. The numerical analysis demonstrates that the memristive chaotic system has infinite coexisting attractors, and a built-in parameter that can achieve global amplitude control of the attractor has also been verified. The omni-directional expansion, coexistence and global amplitude control of the attractor illustrates the extreme multi-stability of the designed system. The analog circuit of the memristive chaotic system has been implemented to verify its feasibility and practicability. Finally, a weak signal detection system based on the grid-scroll memristive chaotic system is constructed. This system not only has excellent robustness to various noise, but also is more sensitive to weak periodic signals. The detection signal-to-noise ratio (SNR) threshold reaches -63 dB. The applicability of this method in practical engineering is verified through the detection of ship radiation signals submerged in ocean background noise.

Chaos suppression of a Seven-dimensional power system based on the predefined-time sliding mode control method

Abstract To suppress chaotic oscillation in the seven-dimensional power system model, the predefined-time sliding mode control of disturbed seven-dimensional chaotic power system is studied in this paper. Firstly, the dynamic characteristics of the seven-dimensional power system are analyzed, and the existence of chaotic attractors is demonstrated. Then, the seven-dimensional power system is divided into three subsystems according to control outputs, and the chaotic control of the seven-dimensional power system is decomposed into the control problem of three subsystems. For each subsystem, a predefined-time sliding mode controller is designed to regulate the system states converge to the desired values. Compared with other existing control methods, the proposed predefined-time sliding mode controller can control the chaotic power system to the expected states at any predefined time. The minimum upper bound of the convergence time is determined by the adjustable parameters of this controller, which make it meet the control requirements conveniently. Finally, the effective and excellent control performance of the predefined-time control scheme is proved by numerical simulation.

Numerical investigation of fractional order SEIR models with newborn immunization using Vieta–Fibonacci wavelets

In this article, we investigate the dynamics of a fractional-order SEIR epidemic model with special emphasis on the vaccination of newborns. By incorporating vaccination directly into the SEIR framework, newborns bypass the susceptible stage and enter the immune class directly, which enhances herd immunity and contributes to the overall reduction in disease spread. A novel operational matrix method based on Vieta–Fibonacci wavelets is developed to approximate the fractional-order SEIR model that includes newborn immunization, where the fractional derivative is taken in the Caputo sense. To begin with, the operational matrices of fractional-order integration are obtained via block-pulse functions. These matrices convert the underlying model into a system of algebraic equations that can solved using any classical method, such as Newton’s iterative method, Broyden’s method, or fsolve command in MATLAB software. The Haar wavelet method is also discussed to show its applicability and efficiency. The obtained results lucidly illustrate the dynamics of susceptible, exposed, infected, and recovered populations during an infectious outbreak. The decline in susceptible and infected individuals reflects the disease’s progression, while vaccination significantly reduces infection peaks. Variations in the fractional parameter α and transmission factor β reveal the influence of these variables on the disease outbreak, with higher values of β leading to rapid transmission. The chaotic attractors of the fractional-order SEIR epidemic model with newborn immunization are graphically represented using Vieta–Fibonacci wavelets.

Physics-inspired machine learning detects ‘unknown unknowns’ in networks: discovering network boundaries from observable dynamics

Abstract Dynamics on networks is often only partially observable in experiment, with many nodes being inaccessible or indeed the existence and properties of a larger unobserved network being unknown. This limits our ability to reconstruct the topology of the network and the strength of the interactions among even the observed nodes. Here, we show how machine learning inspired by physics can be utilized on noisy time series of such partially observed networks to determine which nodes of the observed part of a network form its boundary, i.e. have significant interactions with the unobserved part. This opens a route to reliable network reconstruction. We develop the method for arbitrary network dynamics and topologies and demonstrate it on a broad range of dynamics including non- linear coupled oscillators and chaotic attractors. Beyond these we focus in particular on biochemical reaction networks, where we apply the approach to the dynamics of the Epidermal Growth Factor Receptor (EGFR) network and show that it works even for substantial noise levels.

Swampland conjectures constraints on dark energy from a highly curved field space

We study the interplay of the trans-Planckian censorship conjecture (TCC) and the swampland distance conjecture (SDC) in the context of multifield dark energy in a curved field space. In this scenario, the phase of accelerated expansion is realized as nongeodesic motion in a highly curved field space, reminiscent of models developed in the context of inflation. The model features a stable attractor solution with near constant equation of state w≃−1, and predicts that the current era of accelerated expansion is eternal. The latter implies an eventual conflict with the TCC, which holds that the duration of any epoch of cosmic acceleration is bounded by the requirement that the large-scale observable universe is blind to Planck-scale early universe physics. This tension can be resolved by an interplay with the distance conjecture: for suitable parameter values, the apparent violation of the TCC occurs well after the fields have traversed a Planckian distance. The SDC then predicts a breakdown of the effective field theory before the TCC can be violated. We derive the constraints on the model arising from the SDC+TCC and the de Sitter conjecture. We demonstrate that the model can be consistent with both swampland conjectures and observational data from Planck 2018 and the Dark Energy Spectroscopic Instrument. Published by the American Physical Society 2024

Dynamical systems-inspired machine learning methods for drought prediction

Drought is a naturally occurring phenomenon that affects millions of people and results in billions of dollars in damages each year, with impacts expected to worsen due to climate change. At the same time, definitions of drought are nebulous, and extant quantitative drought indicators suffer from short prediction horizons. One such indicator is the Normalized Vegetation Difference Index (NDVI), which measures photosynthetic activity, making it a strong proxy for vegetation density. Recent studies have identified chaotic attractors in satellite-derived NDVI time-series, suggesting a dynamical systems framework may be helpful for time-series prediction of NDVI. In this study, we compare the performance of a mechanistic model and two physics-informed machine learning methods (Sparse Identification of Nonlinear Dynamics [SINDy] and reservoir computing) on the prediction of NDVI time-series data in drought-prone regions of Kenya. We find that SINDy, a sparse polynomial modelling architecture, narrowly outperforms the other two methods with the use of precipitation data, while also retaining some of the interpretability of the mechanistic model. We also find that none of the methods perform as well in the regions in which the chaotic NDVI attractors were originally identified. We conclude by proposing more sophisticated extensions to the methods presented here, both with and without the availability of precipitation data, that draw on the existing dynamical systems and machine learning literature to enable better quantitative predictions of key drought indicators.

Construction and analysis of symmetric Sprott B multi-attractors with electric implementation

Abstract In chaos theory, a number of systems and models which apparently contain simple ordinary differential equations (ODEs) turn out to show a dynamic characterized by complicated behaviors and complex trajectories. One of such systems is the Sprott B model. We construct some set of multi-attractors based on the Sprott B model where additional parameters and operators are considered. After summarizing important preliminaries relevant to simple chaotic differential systems, the model is firstly solved analytically and numerically, then graphical simulations are provided. The later show coexistence and evolution of two chaotic attractors in a symmetrical representation. Lastly, similar results and expected outcomes are recovered via an electrical circuit implementation, realized using the Field Programmable Gate Array (FPGA) board, the Digital-to-Analog Converter (DAC) and the Rigol Oscilloscope. They also show progressing sets of coexisting multi-attractors.

A novel Chua's based 2-D chaotic system and its performance analysis in cryptography.

In this study, the chaotic behavior of a second-order circuit comprising a nonlinear resistor and Chua's diode is investigated. This circuit, which includes a nonlinear capacitor and resistor among its components, is considered one of the simplest nonautonomous circuits. The research explores various oscillator characteristics, emphasizing their chaotic properties through bifurcations, Lyapunov exponents, periodicity, local Lyapunov region, and resonance. The system exhibits both stable equilibrium points and a chaotic attractor. Additionally, the second objective of this study is to develop a novel cryptographic technique by incorporating the designed circuit into the S-box method. The evaluation results suggest that this approach is suitable for secure cryptographic applications, providing insights into constructing a cryptosystem for images and text based on its complex behavior. Real-life data were analyzed using various statistical and performance criteria after applying the proposed methodology. These findings enhance the reliability of the cryptosystems. Moreover, The proposed methods are assessed using a range of statistical and performance metrics after testing the text and images. The cryptographic results are compared with existing techniques, reinforcing both the developed cryptosystem and the performance analysis of the chaotic circuit.

Jacobi stability, Hamilton energy and the route to hidden attractors in the 3D Jerk systems with unique Lyapunov stable equilibrium

This paper is devoted to reveal the generation mechanism of hidden attractors of the 3D Jerk systems with unique Lyapunov stable equilibrium. In the light of the deviation curvature tensor, the two-parameter regions with Lyapunov stable but Jacobi unstable equilibrium are identified. Within these regions, the system’s dynamics transition from Lyapunov stable but Jacobi unstable equilibrium to hidden periodic and then to hidden chaotic attractors, which the corresponding Hamilton energy tend to be constant, regular and irregular oscillations, respectively. The route to hidden attractors of the systems with Jacobi unstable equilibrium is analyzed under one parameter variation. The results show that the systems initially undergo a subcritical Hopf bifurcation, resulting in a Lyapunov unstable limit cycle, followed by a saddle–node bifurcation of limit cycle, ultimately entering hidden chaotic attractors via the Feigenbaum period-doubling route.

A novel discrete memristive hyperchaotic map with multi-layer differentiation, multi-amplitude modulation, and multi-offset boosting.

In recent years, the introduction of memristors in discrete chaotic map has attracted much attention due to its enhancement of the complexity and controllability of chaotic maps, especially in the fields of secure communication and random number generation, which have shown promising applications. In this work, a three-dimensional discrete memristive hyperchaotic map (3D-DMCHM) based on cosine memristor is constructed. First, we analyze the fixed points of the map and their stability, showing that the map can either have a linear fixed point or none at all, and the stability depends on the parameters and initial state of the map. Then, phase diagrams, bifurcation diagrams, Lyapunov exponents, timing diagrams, and attractor basins are used to analyze the complex dynamical behaviors of the 3D-DMCHM, revealing that the 3D-DMCHM enters into a chaotic state through a period-doubling bifurcation path, and some special dynamical phenomena such as multi-layer differentiation, multi-amplitude control, and offset boosting behaviors are also observed. In particular, with the change of memristor initial conditions, there exists an offset that only homogeneous hidden chaotic attractors or a mixed state offset with coexistence of point attractors and chaotic attractors. Finally, we confirmed the high complexity of 3D-DMCHM through complexity tests and successfully implemented it using a digital signal processing circuit, demonstrating its hardware feasibility.

Hidden multi-scroll and coexisting self-excited attractors in optical injection semiconductor laser system: Its electronic control.

In this paper, we investigate hidden and coexisting self-excited multi-scroll attractors by using a modified rate equations model of semiconductor lasers (REM-SCLs) subjected to optical injection by exploring various quantifying analytical and numerical methods. The multi-leveled dynamics sticks out the existence of several sets of equilibria that asymptotically attract trajectories originating outside of them. Chaos topology based on the impact of equilibria allows the describing of the so-called stable or unstable multi-scroll chaotic attractors. Shaping of the new coexisting self-excited multi-scroll attractor, whose source is from coupling of equilibria, is analyzed, as well as its structural dynamics along with the dynamical emergence of the hidden multi-scroll attractor in the restricted interval, defined by an additional decisive parameter. Additionally, specific 3D plots with embedded contour plots obtained by harnessing two-parameter bifurcation analysis clarify structural dynamics of such a multi-scroll attractor and accurately circumscribe stretching of its fractal-like basin of attraction. Strange metamorphoses undergone by the fractal-like basin of attraction of the studied multi-scroll attractor are stepwisely parsed in the map of two-codimension bifurcation as its scroll number evolves. At last, an electronic circuit of equivalent REM-SCLs is designed and simulated in the PSpice environment alongside a tailored electronic controller. The achieved results align with the ones of numerical analysis; besides, temporal controlling of optical waves pertaining thereto is also fulfilled.

Templex-based dynamical units for a taxonomy of chaos.

Discriminating different types of chaos is still a very challenging topic, even for dissipative three-dimensional systems for which the most advanced tool is the template. Nevertheless, getting a template is, by definition, limited to three-dimensional objects based on knot theory. To deal with higher-dimensional chaos, we recently introduced the templex combining a flow-oriented BraMAH cell complex and a directed graph (a digraph). There is no dimensional limitation in the concept of templex. Here, we show that a templex can be automatically reduced into a "minimal" form to provide a comprehensive and synthetic view of the main properties of chaotic attractors. This reduction allows for the development of a taxonomy of chaos in terms of two elementary units: the oscillating unit (O-unit) and the switching unit (S-unit). We apply this approach to various well-known attractors (Rössler, Lorenz, and Burke-Shaw) as well as a non-trivial four-dimensional attractor. A case of toroidal chaos (Deng) is also treated.

A New Class of Circulant Chaotic Systems with Unidirectional or Mutual Coupling: Theoretical Investigation and Arduino-Based Electronic Implementation

The study of chaotic systems with special properties addresses one of the ongoing research topics. In this paper, we introduce a new class of third-order chaotic systems possessing a cyclic symmetry obeying unidirectional or mutual coupling. These new models are built by considering nonlinearities with three, four, or five segments and are distinguished from each other by the topological structure of the associated chaotic attractor. The projection of the novel chaotic attractors on certain planes reveals their multiscroll or multiwing character. For illustration, one of these models is studied in detail using analytical and numerical methods. The mechanism giving rise to the establishment of the chaotic regime starting from the state of equilibrium under the variation of a parameter is described using bifurcation diagrams, phase portraits, basins of attraction as well as the maximum Lyapunov exponent. Several zones of parameters are identified where the model experiences two or more attractors (i.e. multistability). In order to demonstrate the practical feasibility of the proposed circulant models, a series of experimental measurements are carried out using a physical implementation with an Arduino microcontroller.

Symbolic dynamics approach to find periodic windows: The case study of the Rössler system

Modification of a parameter of a chaotic system may lead to the emergence of a periodic attractor. Under certain assumptions periodic windows (regions in the parameter space in which a periodic attractor exists) densely fill a chaotic region. Usually it is very difficult to prove this property. In this work, we propose a systematic procedure to locate and prove the existence of periodic windows. The method combines the symbolic dynamics based approach to find unstable periodic orbits (UPOs), the continuation method to locate periodic windows (PWs), and interval arithmetic tools to prove their existence. The proposed method is applied to the Rössler system. The existence of several thousands of PWs close to the classical parameter values is proved and periodic attractors very close in the parameter space to the classical Rössler attractor are found. Estimates of measures of sets of parameters for which a periodic attractor exists are calculated.

Dynamic Analysis and FPGA Implementation of Fractional-Order Hopfield Networks with Memristive Synapse

Memristors have become important components in artificial synapses due to their ability to emulate the information transmission and memory functions of biological synapses. Unlike their biological counterparts, which adjust synaptic weights, memristor-based artificial synapses operate by altering conductance or resistance, making them useful for enhancing the processing capacity and storage capabilities of neural networks. When integrated into systems like Hopfield neural networks, memristors enable the study of complex dynamic behaviors, such as chaos and multistability. Moreover, fractional calculus is significant for their ability to model memory effects, enabling more accurate simulations of complex systems. Fractional-order Hopfield networks, in particular, exhibit chaotic and multistable behaviors not found in integer-order models. By combining memristors with fractional-order Hopfield neural networks, these systems offer the possibility of investigating different dynamic phenomena in artificial neural networks. This study investigates the dynamical behavior of a fractional-order Hopfield neural network (HNN) incorporating a memristor with a piecewise segment function in one of its synapses, highlighting the impact of fractional-order derivatives and memristive synapses on the stability, robustness, and dynamic complexity of the system. Using a network of four neurons as a case study, it is demonstrated that the memristive fractional-order HNN exhibits multistability, coexisting chaotic attractors, and coexisting limit cycles. Through spectral entropy analysis, the regions in the initial condition space that display varying degrees of complexity are mapped, highlighting those areas where the chaotic series approach a pseudo-random sequence of numbers. Finally, the proposed fractional-order memristive HNN is implemented on a Field-Programmable Gate Array (FPGA), demonstrating the feasibility of real-time hardware realization.

Three-Dimensional Space Chaotic Scrolls Generated by Memristive Hopfield Neural Network and Its Application in Robot Path Planning

Abstract In this paper, a novel three-dimensional space-scroll chaotic attractor is constructed by introducing piece-wise linear memristors into a three-neurons-based Hopfield neural network to simulate the electromagnetic influence on neural dynamics. The dynamical behaviors of the novel memristive Hopfield neural network are discussed by means of numerical methods including the Lyapunov exponent spectrum, bifurcation diagrams, phase plots and dynamic maps. Furthermore, we also investigated different neurons subject to the memristive electromagnetic influence, where the two-dimensional grid-scroll chaotic attractors can be generated by one neuron subjects to the electromagnetic influence and three-dimensional space-scroll chaotic attractors can be generated by all of the neurons subject to the memristive electromagnetic influence. Moreover, we present a three-dimensional space-scroll scroll attractor-based chaotic mobile robot system. Superiority of the proposed space-scroll-based chaotic mobile robot system compared to existing methods has been demonstrated through numerical simulations.

Dynamical signatures of discontinuous phase transitions: How phase coexistence determines exponential versus power-law scaling.

There are conflicting reports in the literature regarding the finite-size scaling of the Liouvillian gap and dynamical fluctuations at discontinuous phase transitions, with various studies reporting either exponential or power-law behavior. We clarify this issue by employing large deviation theory. We distinguish two distinct classes of discontinuous phase transitions that have different dynamical properties. The first class is associated with phase coexistence, i.e., the presence of multiple stable attractors of the system dynamics (e.g., local minima of the free-energy functional) in a finite phase diagram region around the phase transition point. In that case, one observes asymptotic exponential scaling related to stochastic switching between attractors (though the onset of exponential scaling may sometimes occur for very large system sizes). In the second class, there is no phase coexistence away from the phase transition point, while at the phase transition point itself there are infinitely many attractors. In that case, one observes power-law scaling related to the diffusive nature of the system relaxation to the stationary state.