Three-dimensional Autonomous System Research Articles

The Evolution of the Phase Space Structure Along Pitchfork and Period-Doubling Bifurcations in a 3D-Galactic Bar Potential

We investigate how the phase space structure of a Three-Dimensional (3D) autonomous Hamiltonian system evolves across a series of successive Two-Dimensional (2D) and 3D pitchfork and period-doubling bifurcations, as the transition of the parent families of Periodic Orbits (POs) from stability to simple instability leads to the creation of new stable POs. Our research illustrates the consecutive alterations in the phase space structure near POs as the stability of the main family of POs changes. This process gives rise to new families of POs within the system, either maintaining the same or exhibiting higher multiplicity compared to their parent families. Tracking such a phase space transformation is challenging in a 3D system. By utilizing the color and rotation technique to visualize the Four-Dimensional (4D) Poincaré surfaces of section of the system, i.e. projecting them onto a 3D subspace and employing color to represent the fourth dimension, we can identify distinct structural patterns. Perturbations of parent and bifurcating stable POs result in the creation of tori characterized by a smooth color variation on their surface. Furthermore, perturbations of simple unstable parent POs beyond the bifurcation point, which lead to the birth of new stable families of POs, result in the formation of figure-8 structures of smooth color variations. These figure-8 formations surround well-shaped tori around the bifurcated stable POs, losing their well-defined forms for energies further away from the bifurcation point. We also observe that even slight perturbations of highly unstable POs create a cloud of mixed color points, which rapidly move away from the location of the PO. Our study introduces, for the first time, a systematic visualization of 4D surfaces of section within the vicinity of higher multiplicity POs. It elucidates how, in these cases, the coexistence of regular and chaotic orbits contributes to shaping the phase space landscape.

Kosambi-Cartan-Chern perspective on chaos: Unveiling hidden attractors in nonlinear autonomous systems.

This article confronts the formidable task of exploring chaos within hidden attractors in nonlinear three-dimensional autonomous systems, highlighting the lack of established analytical and numerical methodologies for such investigations. As the basin of attraction does not touch the unstable manifold, there are no straightforward numerical processes to detect those attractors and one has to implement special numerical and analytical strategies. In this article we present an alternative approach that allows us to predict the basin of attraction associated with hidden attractors, overcoming the existing limitations. The method discussed here is based on the Kosambi-Cartan-Chern theory which enables us to conduct a comprehensive theoretical analysis by means of evaluating geometric invariants and instability exponents, thereby delineating the regions encompassing chaotic and periodic zones. Our analytical predictions are thoroughly validated by numerical results.

Generating Multiwing Hidden Chaotic Attractors With Only Stable Node-Foci: Analysis, Implementation, and Application

Based on Shil'nikov criterion, i.e., extending the number of unstable saddle-focus -type equilibria of chaotic system, a rich and varied multiwing chaos with complicated topology and remarkable pseudorandomness has been coined. However, it is a challenging task to construct multiwing chaos with only stable node-focus -type equilibria as the Shil'nikov criterion is not applicable in the circumstances. To eliminate this difficulty and enrich the multiwing chaos, we construct a simple three-dimensional autonomous system with only two symmetric stable equilibria, and further introduce a sawtooth wave modulation function to configure more equilibria. The highlight is that multiwing chaotic attractors can be generated from system with two or multiple stable node-foci. This characteristic indicates that the attractors can be hidden oscillating and are difficult to be located in the small neighborhoods of equilibria, which makes the system has good concealment performance for security applications. The design mechanism and system properties are theoretically analyzed and numerically investigated using phase portraits, finite-time Lyapunov exponents, and basins of attraction. Hardware experiments based on field-programmable gate array verify the feasibility of the system. Finally, based on the generated multiwing hidden chaotic attractors, an image encryption scheme with high security performance is designed to enhance its application.

Jacobi Stability for T-System

In this paper will be considered a three-dimensional autonomous quadratic polynomial system of first-order differential equations with three real parameters, the so-called T-system. This system is symmetric relative to the Oz-axis and represents a special type of the generalized Lorenz system. The approach of this work will consist of the study of the nonlinear dynamics of this system through the Kosambi–Cartan–Chern (KCC) geometric theory. More exactly, we will focus on the associated system of second-order differential equations (SODE) from the point of view of Jacobi stability by determining the five invariants of the KCC theory. These invariants determine the internal geometrical characteristics of the system, and particularly, the deviation curvature tensor is decisive for Jacobi stability. Furthermore, we will look for necessary and sufficient conditions that the system parameters must satisfy in order to have Jacobi stability for every equilibrium point.

Cascade tri-neuron hopfield neural network: Dynamical analysis and analog circuit implementation

This paper studies a cascade tri-neuron Hopfield neural network (CTN-HNN) with no connection between the first neuron and the third neuron. Such incompletely connected neuronal structure may be part of complex neural networks with negligible inputs, and may be present in abnormal neural networks with some neurological diseases. It was proved that the stable point, limit cycle, single-scroll chaotic attractor and double-scroll chaotic attractor were observed in the network by adjusting the synaptic weight from the second neuron to itself. To further dynamic analysis, we first demonstrate that this three-dimensional (3D) autonomous nonlinear dynamical system is symmetric about the origin and bounded ultimately, and has one unstable saddle-focus zero point and two symmetric nonzero points with different stability types. Afterwards, taking two synaptic weights as adjustable parameters, the numerical analysis reveals that the CTN-HNN exhibits coexisting chaotic and periodic attractors, coexisting bursting firings and long-term transient chaos by employing multiple numerical methods. Finally, an analog printed circuit is built using one main circuit module and three activation function modules, thus effectively demonstrating the complex dynamics of the network.

A Chen-Like Model: High Periodicity Leading to Chaotic Dynamics

The Chen circuit system, a three-dimensional autonomous system with intriguing dynamics, has gained considerable attention due to its potential applications in diverse scientific fields. In this article, a Chen-like system is defined and a comprehensive dynamics studied. The system exhibits chaotic dynamics, limit cycles, and bifurcations, making it a captivating subject of study. By employing numerical simulations and analytical techniques, we explore the system’s stability and identify critical parameter values leading to qualitative changes. Notably, we delve into Hopf bifurcations, which give rise to stability changes and the emergence of limit cycles. Furthermore, we analyze the fractal dimension of the system’s attractor, providing insights into its complexity and self-similarity. Through a systematic examination of the Chen-like system, we deepen our understanding of its intricate dynamics and offer valuable insights into the underlying mechanisms. The findings contribute to the field of dynamical systems and hold potential implications in areas such as chaos-based secure communications, signal processing, and nonlinear control. This work serves as a valuable reference for researchers and practitioners interested in the dynamics and bifurcation analysis of nonlinear systems.

Three-Dimensional Drone Exploration with Saliency Prediction in Real Unknown Environments

In this paper, we propose a three-dimensional autonomous drone exploration system (ADES) with a lightweight and low-latency saliency prediction model to explore unknown environments. Several studies have applied saliency prediction in drone exploration. However, these studies are not sufficiently mature. For example, the computational complexity and the size of the developed prediction models have not been considered. In addition, some studies have only proposed saliency prediction models without actually applying them to drones. The ADES system proposed in this paper has a small and fast saliency prediction model and uses a novel drone exploration approach based on visual-inertial odometry to solve the practical problems encountered during drone exploration, such as collisions with and the repeated exploration of salient objects. The proposed ADES system performs comparably to the state-of-the-art, multiple-discontinuous-image saliency prediction network TA-MSNet and enables drones to explore unknown environments with high efficiency.

Microcontroller Implementation, Controls, and Synchronization of Three-Dimensional Autonomous System with a Parabolic Equilibrium Point

The microcontroller implementation, controls, and synchronization of a three-dimensional (3D) autonomous system with a parabolic equilibrium point are investigated in this paper. The system in question displays a reverse period doubling route hidden chaotic attractors with two different shapes. Then, the partial and total amplitude controls of the system are achieved by inserting two parameters. A microcontroller implementation is performed in order to confirm the results obtained from the numerical simulations. It is found that the results from the numerical simulations and microcontroller implementation qualitatively agree with each other. The sliding mode controllers are designed to control chaos in the system under study. With the sliding mode control method, the numerical simulations confirm that chaos can be controlled in the 3D autonomous system with a parabolic equilibrium point. In addition, two chaotic 3D autonomous systems with a parabolic equilibrium point and the same parameters are synchronized by the use of a unidirectional linear error feedback coupling scheme. Finally, an active control technique is applied to bring about chaos synchronization between two chaotic 3D autonomous systems with a parabolic equilibrium and different parameters.

Time-reversal, tori families,\query{Q1} and canards in the Sprott A and NE9 systems.

Quadratic three-dimensional autonomous systems may display complex behavior. Studying the systems Sprott A and NE9, we find families of tori and periodic solutions both involving canards. Using time-reversal and symmetry, we are able to explain in these two systems both the analysis and origin of tori, periodic solutions, and the numerics of these objects. For system NE9, unbounded solutions exist that admit analytic description by singular perturbation theory of the flow near infinity, also we observe torus destruction and a new chaotic attractor (Kaplan-Yorke dimension 2.1544) produced by a period-doubling scenario. The subtle numerics of periodic solutions involving canards is explained in the final section.

A dilaton-axion model for string cosmology

The generic scale-invariant theory of an axion and a dilaton coupled to gravity in d-dimensions is generalized to a ‘universal’ one-axion model with two dilatons that reproduces itself under consistent dimensional-reduction/truncation. Flat FLRW cosmologies are shown to correspond to trajectories of a three-dimensional autonomous dynamical system, which we analyse with a focus on accelerated cosmic expansion, deriving the precise swampland bounds that exclude eternal acceleration. We also show that for two sets of values of its three independent parameters, the model is a consistent truncation of maximal ‘massive’ supergravity theories arising from string/M-theory; for these maximal-supergravity parameter values the FLRW cosmologies include some with a transient de Sitter-like phase, but not the recurring de Sitter-like phase or eternal cosmic acceleration that is possible for other parameter values.

Analysis of the Stability and Hopf Bifurcation of a Three-Dimensional System with Delays

We propose a model of three-dimensional autonomous system with delays. We explore the dynamical behavior of the proposed autonomous system by examining bifurcation diagrams, Lyapunov exponents, equilibrium and stability, and the influence of time delay on Hopf bifurcation. A bifurcation theory is used to analyze and detail the problem. In addition, the explicit algorithm that determines the direction of Hopf bifurcation, along with the stability of bifurcating periodic, has been established. Also, there are specific operating conditions that must be met in order to achieve Hopf bifurcation. In the proposed autonomous system, we analyze the procedures for designing chaotic based systems including parameter selection, discretization of the results, as well as exploring the changing regularity of the bifurcation value. A series of numerical simulations is presented to illustrate the analytical results.

Rocard’s 1941 Chaotic Relaxation Econometric Oscillator

In the beginning of the Second World War, the French physicist, Yves Rocard, published a book entitled Théorie des Oscillateurs (Theory of Oscillators). In Chapter V, he designed a mathematical model consisting of a set of three nonlinear differential equations and allowing to account for economic cycles. Numerical integration of his model has highlighted a chaotic attractor. Its analysis with classical tools such as bifurcation diagram and Lyapunov Characteristic Exponents has confirmed the chaotic features of its solution. It follows that Rocard’s 1941 chaotic econometric model has thus most likely preceded Lorenz’ butterfly of twenty-two years. Moreover, apart from this historical discovery which upsets historiography, it is also established that this “new old” three-dimensional autonomous dynamical system is a new jerk system whose solution exhibits a chaotic attractor the topology of which varies, from a double scroll attractor to a Möbius-strip and then to a toroidal attractor, according to the values of a control parameter.

Dynamical Behavior of a New Chaotic System with One Stable Equilibrium

This paper reports a simple three-dimensional autonomous system with a single stable node equilibrium. The system has a constant controller which adjusts the dynamic of the system. It is revealed that the system exhibits both chaotic and non-chaotic dynamics. Moreover, chaotic or periodic attractors coexist with a single stable equilibrium for some control parameter based on initial conditions. The system dynamics are studied by analyzing bifurcation diagrams, Lyapunov exponents, and basins of attractions. Beyond a fixed-point analysis, a new analysis known as connecting curves is provided. These curves are one-dimensional sets of the points that are more informative than fixed points. These curves are the skeleton of the system, which shows the direction of flow evolution.

Topological classification of periodic orbits in the generalized Lorenz-type system with diverse symbolic dynamics

Periodic orbits play important roles in the analysis of dynamic behaviors in chaotic systems, and they are fundamental keys to understanding the properties of the strange attractor. In this paper, the unstable periodic orbits of a three-dimensional (3D) autonomous Lorenz-type system, the so-called generalized Lorenz-type system (GLTS), are investigated using the variational method. Taking the parameters of the GLTS as (a,b,c)=(10,100,10.4), we use two cycles as basic building blocks to establish 1D symbolic dynamics, and all short unstable cycles up to topological length 5 are found. With typical parameters (a,b,c)=(10,42.72,1), the correlation between the Burke-Shaw system (BSS) and the GLTS is discussed, symbolic dynamics for four letters are identified, and the periodic orbits are labeled inside or outside of the attractor based on whether the symbol sequence of the cycle contains the building blocks with the self-linking number 1. The variational method verifies the effectiveness of locating the periodic orbits, and topological classification provides a novel way to build appropriate symbolic dynamics. We also utilize the homotopy evolution approach to continuously deform the found periodic orbits while varying different parameters, and analyze the various bifurcations in the GLTS. The current work further expands the approach of establishing symbolic dynamics while analyzing the periodic orbits in chaotic systems, and can be applied to the study of turbulence and other complex problems.

Dynamical equilibrium states of a class of irrotational non-orthogonally transitive G 2 cosmologies: II. Models with one hypersurface-orthogonal Killing vector field

We consider a class of inhomogeneous self-similar cosmological models in which the perfect fluid flow is tangential to the orbits of a three-parameter similarity group. It is assumed throughout that the source is a perfect fluid with linear equation of state. We restrict the similarity group to possess both an Abelian G 2, and a single hypersurface orthogonal Killing vector field, and we restrict the fluid flow to be orthogonal to the orbits of the Abelian G 2. The temporal evolution of the models is forced to be power law, due to the similarity group, and the Einstein field equations reduce to a three-dimensional autonomous system of ordinary differential equations which is qualitatively analysed in order to determine the spatial structure of the models. The existence of two classes of well-behaved models is demonstrated. The first of these is asymptotically spatially homogeneous and matter dominated, and the second is vacuum dominated and either asymptotically spatially homogeneous or acceleration dominated, at large spatial distances.

A physical memristor based Muthuswamy\u2013Chua\u2013Ginoux system

In 1976, Leon Chua showed that a thermistor can be modeled as a memristive device. Starting from this statement we designed a circuit that has four circuit elements: a linear passive inductor, a linear passive capacitor, a nonlinear resistor and a thermistor, that is, a nonlinear “locally active” memristor. Thus, the purpose of this work was to use a physical memristor, the thermistor, in a Muthuswamy–Chua chaotic system (circuit) instead of memristor emulators. Such circuit has been modeled by a new three-dimensional autonomous dynamical system exhibiting very particular properties such as the transition from torus breakdown to chaos. Then, mathematical analysis and detailed numerical investigations have enabled to establish that such a transition corresponds to the so-called route to Shilnikov spiral chaos but gives rise to a “double spiral attractor”.

Energy determines multiple stability in time-delayed systems

Infinite dimensions always challenge the analysis of multiple stability in nonlinear time-delayed systems, as the computation and visualization of conventional basin of attraction are hampered by the increase in systems’ dimensions. To address this issue, this paper introduces an orthonormal basis to approximate the delayed states, uses their signal energy to represent them, and generalises the concept of basin of attraction into stochastic, where each pixel of the basin has the same energy level for each delayed state but corresponds to many initial conditions. Thus, the probabilities are estimated by Monte Carlo method, which is then extensively boosted by artificial neural networks including both classification and regression types. This procedure has been successively applied in the analysis of multiple stability in three typical time-delayed systems, which are a two-dimensional autonomous cutting process, a three-dimensional autonomous neural system, and a two-dimensional non-autonomous forced vibration isolator. They, respectively, have one, two, and two delayed states, with two, three, and five coexisting attractors. It is found that the energy distribution in the delayed state determines both the convergence of Monte Carlo simulation and sensitivity of the classification neural network. It is also seen that the performance of classification neural networks decreases with respect to the increase in the number of attractors, but the regression neural networks show a robuster performance. As a result, the stochastic basin of attraction can be accurately and efficiently computed to reveal the multiple stability in various time-delayed systems.

Completely integrable dynamical systems of Hopf–Langford type

In this work we consider a three-dimensional autonomous system of nonlinear ordinary differential equations, which may be thought of as a generalization of the well-known Hopf–Langford system introduced about forty years ago. This dynamical system turned out to be equivalent to the nonlinear force-free Duffing oscillator. In three special cases, it is found to be completely integrable. To the best of our knowledge, these facts have not been noticed so far in the rich literature on the subject. In the aforementioned three special cases, the general solutions of the respective systems are expressed in explicit analytical form by means of elementary and Jacobi elliptic functions depending on the values of the system parameters. This allowed us to characterize in details the dynamics of the regarded systems.

Coexistence of multi-scroll chaotic attractors for fractional systems with exponential law and non-singular kernel

Abstract In this paper, we present mathematical analysis and numerical simulation of a three-dimensional autonomous fractional system with coexistence of multi-scroll chaotic attractors. We replaced the classical derivatives of such system with the Caputo-Fabrizio fractional derivative. This derivative combines both the exponential laws and non-singular kernels in its formulation which makes it special and useful. A two-step Adams-Bashforth scheme is derived for the approximation of the fractional derivative with exponential law and non-singular kernel. We then presented both numerical results and graphical results by considering many values of the fractional-order parameter β ∈ (0, ]. We demonstrate that the observed chaotic behavior conduct perseveres as the fractional-order parameter approaches 1.

Hopf bifurcation, offset boosting and remerging Feigenbaum trees in an autonomous chaotic system with exponential nonlinearity

The present paper executes an analysis of a relatively simple chaotic system with exponential nonlinearity. This dimensionless dynamic system has been obtained from the chaotic circuit introduced by Ma and collaborators (nonlinear Dyn. https://doi.org/10.1007/s11071-018-4307-x ) using scale transformation. The major dynamic properties of this model have been investigated, in particular, the equilibrium points and the dissipation property of the model explored. Based on usual diagnostic tools such as phase portraits, bifurcation diagrams, frequency spectrum, Lyapunov spectrum and basins of attraction, the nonlinear behaviours of the model have been highlighted. We demonstrated that the three-dimensional autonomous system experiences the especial feature of subcritical Hopf bifurcation (when the control parameter exceeds a critical value); equally the period-doubling bifurcation, saddle-node bifurcation, offset boosting, reverse period doubling and antimonotonicity have been found. With the use of the normal form theory (Hassard algorithm), a formula for finding the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions is presented. Furthermore, by using the Marsden–MacCracken index, we show that this Hopf bifurcation at equilibria is non-degenerate. At the end, the experimental results are given to further support the theoretical analysis.