Attractor Research Articles

Unstable dimension variability and heterodimensional cycles in the border-collision normal form.

Chaotic attractors commonly contain periodic solutions with unstable manifolds of different dimensions. This allows for a zoo of dynamical phenomena not possible for hyperbolic attractors. The purpose of this Letter is to emphasize the existence of these phenomena in the border-collision normal form. This is a continuous, piecewise-linear family of maps that is physically relevant as it captures the dynamics created in border-collision bifurcations in diverse applications. Since the maps are piecewise linear, they are relatively amenable to an exact analysis. We explicitly identify parameter values for heterodimensional cycles and argue that the existence of heterodimensional cycles between two given saddles can be dense in parameter space. We numerically identify key bifurcations associated with unstable dimension variability by studying a one-parameter subfamily that transitions continuously from where periodic solutions are all saddles to where they are all repellers. This is facilitated by fast and accurate computations of periodic solutions; indeed the piecewise-linear form should provide a useful testbed for further study.

Chaos, periodic structures, and multistability: Complex dynamical behaviors of an eco-epidemiological model in parameter planes.

The environmental characteristics of a biological system are imbibed in some particular parameters of that system. Significant changes in any system parameter exert influence on the system dynamics as well as the persistence of interacting species. In this article, we explore the rich and tangled dynamics of an eco-epidemiological system by studying different parametric planes of the system. In the parameter planes, we find a variety of complex and subtle properties of the system, like the presence of a variety of intricate regular structures within irregular regimes, that cannot be found through a single parameter variation. Also, we find a new type of structure like an "eye" in a parametric plane. We notice the bistability between distinct pairs of attractors and also identify the coexistence of three periodic attractors. The most notable observation of this study is the coexistence of three periodic attractors and a chaotic attractor, which is a rare occurrence in biological systems. We also plot the basins for each set of coexisting attractors and see the existence of fractal basins in the system, which look like a "conch." The appearance of fractal basins in a system causes enormous complications in predicting the system's state in the long run. Variations in initial conditions and changes in parameters in parametric planes are key to managing the behavior of a system.

Влияние деформаций инструмента и заготовки в направлении скорости резания на динамику обработки

Numerous studies were devoted to studying the dynamic cutting system considered in combination of the tool and the workpiece sub-systems interacting through a dynamic connection formed by the machining process. Dynamic coupling is simulated by the cutting forces represented in the system state coordinates, which determines the system properties. Several models were proposed to describe the dynamic connection reflecting various experimentally observed effects and contributing to the self-excitation. Regenerative effect of the tool trace left on the workpiece at the previous revolution was considered. The regenerative effect was studied on the basis of assumption that the retarding argument in the forces description was remaining unchanged. The paper studies the regenerative effect influence on dynamic properties of the cutting process taking into account (unlike the known works) the retarded argument dependence on the tool deformation displacement. This could fundamentally change properties of the system under consideration in the unity of stability and the attracted formed deformation displacement sets (limit cycles, invariant tori and chaotic attractors). Results of the mathematical simulation are presented taking into account the regenerative self-excitation, where the retarded argument is the state coordinates function. Bifurcation diagrams of attracting sets of the deformation displacements are considered, and conditions for formation of their superlow-frequency components of the complex spatial-temporal structure are discussed. Research results are aimed at determining the machining conditions based on requirements for ensuring the specified quality of parts manufacturing using the longitudinal turning example.

Yet New Extreme Multistable Chaotic System

The upsurge of discovering complex dynamics in simple chaotic systems is spring up in academia. This essay is devoted to investigating extreme multistability in a simple chaotic system. Here, only six terms with one nonlinearity form the mathematical model of this system and brings it infinitely many unstable equilibria. The unstable equilibria make the system possess the ability to yield extreme multistability, including various types of chaotic and periodic attractors. The dynamic behaviors, e.g. extreme multistability and total amplitude control are numerically studied. To verify the proposed system’s physical existence, an analog circuit and implementation utilizing the microcontroller is devised as well, where their outcomes perfectly matched with those of simulations.

Predicting chaotic statistics with unstable invariant tori.

It has recently been speculated that long-time average quantities of hyperchaotic dissipative systems may be approximated by weighted sums over unstable invariant tori embedded in the attractor, analogous to equivalent sums over periodic orbits, which are inspired by the rigorous periodic orbit theory and which have shown much promise in fluid dynamics. Using a new numerical method for converging unstable invariant two-tori in a chaotic partial differential equation (PDE), and exploiting symmetry breaking of relative periodic orbits to detect those tori, we identify many quasiperiodic, unstable, invariant two-torus solutions of a modified Kuramoto-Sivashinsky equation. The set of tori covers significant parts of the chaotic attractor and weighted averages of the properties of the tori-with weights computed based on their respective stability eigenvalues-approximate average quantities for the chaotic dynamics. These results are a step toward exploiting higher-dimensional invariant sets to describe general hyperchaotic systems, including dissipative spatiotemporally chaotic PDEs.

Novel Extreme Multistable Tabu Learning Neuron: Circuit Implementation and Application to Cryptography

The complex dynamics of a simple memristive tabu learning neuron are considered in this paper. The analysis of the stability of its equilibria revealed that it displays self-excited dynamics. The investigation of the dynamics of the considered model highlighted that it is extremely sensitive to the initial conditions. That sensitivity to the initial conditions is supported by the coexistence of an infinite number of stable states for the same set of system parameters but using different initial states. Among the infinity of coexisting stable states, there are periodic, quasiperiodic, and chaotic ones. The coexistence of an infinite number of chaotic attractors found in this work and not yet reported in such a model represents the first important contribution of this work. The circuit of the coupled neuron is also realized in the PSPICE simulation environment to further support the obtained result in extreme multistability. Therefore, the revealed chaotic dynamics of the memristive tabu learning neuron is applied to compress and encrypt digital medical images. The compressed sensing approach is combined with DNA coding/decoding to achieve high compression/encryption performances, including very low computational cost (encryption time t=0.162ms, encryption throughput ET=1618.1 MB/s, number of CPU cycles NC=1.8) useful for real-time compression. The compression/encryption method developed in this work represents the second main contribution based on the results of the analysis metrics.

Multi-scroll attractor and its broken coexisting attractors in cyclic memristive neural network

This paper proposes a simple-structured memristive neural network, which incorporates self-connections of memristor synapses alongside both unidirectional and bidirectional connections. Different from other multi-scroll chaotic systems, this network structure has a more concise three-neuron structure. This simple memristive neural network can generate a number of multi-scroll attractors in manageable quantities and shows the characteristics of the coexisting attractors and amplitude control. In particular, when the parameters are changed, the coexisting attractors break up around the center of gravity into two centrosymmetric chaotic attractors. Abundant dynamic behaviors are studied through phase portraits, bifurcation diagrams, Lyapunov exponents, and attraction basins. The feasibility of the system is demonstrated by building a circuit realization platform.

From period-doubling bursting to chaotic–periodic bursting in a modified Chua’s circuit

The aim of this paper is to reveal the dynamical mechanism of bursting oscillations in non-smooth dynamical systems, and the interest is focused on the effects of period-doubling bifurcation and chaotic attractor on the bursting oscillations. By modifying a fourth-order Chua’s circuit, a dynamical system possessing non-smooth switching manifold and multiple scale variables, is established. By taking the slow-varying variable as a bifurcation parameter, Subcritical non-smooth Hopf bifurcation, C-bifurcation, and period-doubling bifurcation are obtained in the fast subsystem. Further more, chaotic attractors, which are generated from a series of period-doubling bifurcations, are also observed. Based on the numerical simulations and bifurcation analysis of the fast subsystem, eight typical bursting patterns are obtained, and their dynamical mechanism is revealed. Study shows that period-2 limit cycle in the fast subsystem may cause the slow–fast system to oscillates in alternating small and large amplitudes; A complex spiking attractors made up of limit cycles and chaotic attractors may lead to spiking states changing from chaotic oscillations to periodic ones, and the trajectory may jump directly from a chaotic attractor to other stable one without undergoing any bifurcation of the fast subsystem.

Applications of the generalized gamma function to a fractional-order biological system

In this work, variety of complex dynamics are found in a fractional-order antimicrobial resistance (AMR) model based on the generalized Gamma function. Firstly, the extended left and right Caputo fractional differential operators, respectively, ELCFDO and ERCFDO are introduced. The basic features of the ELCFDO are outlined. The ELCFDO is shown to have a new fractional parameter that affects the occurrence of the complex dynamics in the fractional AMR system. Secondly, discretization of the ELCFDO is studied using piecewise constant arguments. Then, complex dynamics of the discretized version of the fractional AMR system involving the ELCFDO are investigated such as the existence of Neimark–Sacker (NS) and flip bifurcations, the existence of closed invariant curves (CIC), the existence of strange attractors with fractal or multi-fractal structures, and chaotic attractors. Finally, an extension of the fractal-fractional operator (FFO) that combines fractal and fractional differentiation is carried out based on the generalized Gamma function. The extended FFO (EFFO) is applied to the proposed AMR system, which also generates similar complex dynamics.

A Chaos-Enhanced Fractional-Order Chaotic System with Self-Reproduction Based on a Memcapacitor and Meminductor

Chaotic signals generated by chaotic oscillators based on memory elements are suitable for use in the field of confidential communications because of their very good randomness. But often their maximum Lyapunov exponent is not high enough, so the degree of randomness is not enough. It can be chaos enhanced by transforming it to fractional order using the Caputo differential definition. In this paper, based on the proposed hyperchaotic oscillator, it is extended to a fractional-order form to obtain a chaos-enhanced fractional-order memcapacitor meminductor system, in which several different styles of chaotic and hyperchaotic attractors are found. The dynamical behaviour of the system is studied using bifurcation diagrams, Lyapunov exponent spectrums and Lyapunov dimensions. The multistability of the system is explored in different initial orbits, and the spectral entropy complexity of this system is examined. Finally, a hardware implementation of the memcapacitor meminductor system is given, which demonstrates the effectiveness of the system. This study provides a reference for the study of chaos-enhanced.

Learning strange attractors with reservoir systems

This paper shows that the celebrated embedding theorem of Takens is a particular case of a much more general statement according to which, randomly generated linear state-space representations of generic observations of an invertible dynamical system carry in their wake an embedding of the phase space dynamics into the chosen Euclidean state space. This embedding coincides with a natural generalized synchronization that arises in this setup and that yields a topological conjugacy between the state-space dynamics driven by the generic observations of the dynamical system and the dynamical system itself. This result provides additional tools for the representation, learning, and analysis of chaotic attractors and sheds additional light on the reservoir computing phenomenon that appears in the context of recurrent neural networks.

Nonlinear dynamics, coexistence of attractors and microcontroller implementation of a modified Helmholtz Jerk oscillator

In this work, we converted a two-dimensional modified Helmholtz oscillator into a three-dimensional modified Helmholtz jerk oscillator. The study of the stability of the fixed points is made and by using the theorem of Hopf, the condition of existence of the bifurcation of Hopf is sought. By numerical simulations relating to the diagrams of the basin of parameters, attraction, bifurcation, the Lyapunov exponents and the phase portrait, the global dynamics as well as the coexistence of the attractors of the system are analyzed. This study revealed that the considered modified Jerk Helmholtz oscillator can generate Hopf bifurcation, bistable limit cycles, coexistence of chaotic and periodic attractors for appropriate choices of system parameter values. The microcontroller based implementation of the modified Jerk Helmholtz oscillator is proposed to experimentally verify the obtained analytical and numerical results. Finally, to control the amplitude of the Lyapunov attractor and exponent, we added two new parameters in the modified Helmholtz jerk oscillator.

A general method for generating multi-scroll and multi-wing chaotic systems and its implementation of attractor reproduction

Abstract In comparison to traditional chaotic systems, the multi-scroll and multi-wing chaotic systems are more complicated. The design and execution of sophisticated multi-scroll or multi-wing chaotic attractors attract a lot of attention. However, these constructed nonlinear functions cannot be applied to extended multi-scroll and multi-wing attractors at the same time. To this end, this paper proposes a new function which can be used to generate multi-scroll and multi-wing chaotic attractors in both double-scroll and double-wing chaotic systems. Using this function, multi-scroll and multi-wing chaotic systems can be constructed directly without relying on whether the chaotic system has some symmetry (odd symmetry or even symmetry). The construction method presented is generally applicable to chaotic systems with multi-scroll and multi-wing self-excited attractors.The main point of this method is as follows: firstly, the piecewise linear (PWL) saturation function is nested within the cosine nonlinearity function , and the resulting nested COS-PWL function. Secondly, to enable the expansion of multi-wing and multi-scroll, the nested COS-PWL function is incorporated into the double-wing and double-scroll systems in different manners. The maximum Lyapunov exponent (MLE) and the bifurcation diagram route for increasing the number of wings and scrolls confirm the feasibility and effectiveness of the method. Finally, the three-element method is used to determine a Sinusoidal function, which can generate attractor self-reproduction in the corresponding dimension by replacing the state variables of the multi-scroll and multi-wing systems, so that an infinite number of coexisting attractors can be obtained by simply changing the initial values of the variables, i.e., multiple stability can be generated.

Multi-scroll chaotic attractors with multi-wing via oscillatory potential wells

The oscillatory potential well can cause the mass points in it to move chaotically, which can be considered as a physical mechanism of chaos generation. Based on this physical mechanism, the oscillatory multi-well potential with multi-concave is used to generate controllable multi-scroll chaotic attractors with multi-wing. These attractors have two kinds of topological units: scroll and wing. The topological structure of the attractor depends on the shape of the potential well, that is, the wells and the concaves correspond to the scrolls and wings of the attractor. By constructing potential wells with different shapes, we can obtain attractors with different topological structure. The number of scrolls and wings of the chaotic attractor can be controlled by the oscillation amplitude of the potential well and damping coefficient, that is, stronger oscillation or smaller damping allows the mass points to enter the higher potential wells or concaves, resulting in attractors with more scrolls or wings. As the number of scrolls or wings increases, the attractors become more complex, as expressed by Poincaré maps and quantified by the Kaplan-Yorke dimensions. Kaplan-Yorke dimensions of attractors can be adjusted continuously from 2 to 3 by the two control parameters. Finally, the results are confirmed by Multisim simulation circuit and actual analog circuit.

Construction of a New 2D Hyperchaotic Map with Application in Efficient Pseudo-Random Number Generator Design and Color Image Encryption

This paper proposes a new two-dimensional discrete hyperchaotic system and utilizes it to design a pseudo-random number generator (PRNG) and an efficient color image encryption algorithm. This hyperchaotic system has very complex dynamic properties and can generate highly random chaotic sequences. The complex hyperchaotic characteristics of the system are confirmed via bifurcation diagram, chaotic attractor, Lyapunov exponents, correlation analysis, approximate entropy and permutation entropy. Compared with some traditional discrete chaotic systems, the new chaotic system has a larger range of chaotic parameters and more complex hyperchaotic characteristics, making it more suitable for application in information encryption. The proposed PRNG can generate highly random bit sequences that can fully pass all NIST testing items. The proposed color image encryption algorithm achieves cross-channel permutation and diffusion of pixels in parallel. These strategies not only greatly improve the encryption speed of color images, but also enhance the security level of cipher images. The simulation experiments and security analysis results show that the algorithm has strong robustness against differential attacks, statistical attacks and interference attacks, and has good application potential in real-time secure communication applications of color images.

Analysis of the configuration characteristics of the chaotic attractor for machine tool dynamic error under the singular spectrum

To explore the chaotic characteristics of the motion error of CNC machine tools, this paper proposes a chaotic attractor feature extraction method for the early motion error of CNC machine tools. Firstly, the chaotic verification of each time series feature of machine tool motion error proves that the change of CNC machine tool motion error is a nonlinear change process with chaotic characteristics. It then establishes the multivariate error phase space and performs a singular spectrum analysis. Because the motion error is strongly disturbed by noise, it is difficult to determine the number of singular values by the traditional singular spectrum analysis method, and this paper proposes to calculate the phase space index of the reconstructed sequence by traversing the number of singular values, to determine the main feature components; Finally, under different noise environments and different signal lengths, three feature extraction methods are introduced for experimental comparison with this paper's method, which shows that the chaotic attractor reconstructed after determining the number of singular values has a more apparent geometric structure. The method proposed in this paper has advantages in machine tool motion error.

Lateral nonlinear vibration of the camshaft considering the structural clearance and load force

To reveal the influence of variations in dynamic characteristics on the working performance of the camshaft, the lateral nonlinear vibration of the camshaft considering the structural clearance and load force is studied. By using the Magnus series expansion method, the response of lateral nonlinear vibration is solved, and the numerical computation scheme applied to the global evolution characteristics of attractors and their basins is developed, which solves the problems of inadequate accuracy and long-term numerical unreliability in numerical solutions and simulations. By analyzing the attractors and basins, single-parameter bifurcation, and bi-parameter dynamic characteristics, the influence of different parameters on the nonlinear dynamic characteristics of the camshaft system is revealed. The results show that as the structural clearance increases, the attraction basins of both periodic and chaotic attractors exhibit tongue-shaped fractal features, and the camshaft system changes from the coexistence of two types of period-3 attractors to chaotic attractors, which reveals the mechanism of global vibration instability of the camshaft system. In addition, when the camshaft system evolves from the flutter to forced vibration conditions, the period-doubling bifurcation behavior occurs, and the parameter ranges for stable operation of the system increase. More importantly, the bi-parameter dynamic characteristics confirm that the parameter ranges for the periodic motion of the system are enlarged when the load force increases, and there exist multi-parameter optimal matching solutions for the stable motion of the camshaft system. The research results reveal the evolution mechanism of the lateral nonlinear vibration of the camshaft, which can provide a theoretical reference for suppressing the vibration and improving the working performance.

FPGA-Based Implementation of a New 3-D Multistable Chaotic Jerk System with Two Unstable Balance Points

Mechanical jerk systems have applications in several areas, such as oscillators, microcontrollers, circuits, memristors, encryption, etc. This research manuscript reports a new 3-D chaotic jerk system with two unstable balance points. It is shown that the proposed mechanical jerk system exhibits multistability with coexisting chaotic attractors for the same set of system constants but for different initial states. A bifurcation analysis of the proposed mechanical jerk system is presented to highlight the special properties of the system with respect to the variation of system constants. A field-programmable gate array (FPGA) implementation of the proposed mechanical jerk system is given by synthesizing the discrete equations that are obtained by applying one-step numerical methods. The hardware resources are reduced by performing pipeline operations, and, finally, the paper concludes that the experimental results of the proposed mechanical jerk system using FPGA-based design show good agreement with the MATLAB simulations of the same system.

Stochastic Bifurcations and Multistage Order–Chaos Transitions in a 4D Eco-Epidemiological Model

A tritrophic “prey-intermediate predator-top predator” population system with disease in the intermediate predator is considered. For this 4D-model, the bifurcation analysis is performed. In this analysis, the rate of the disease transmission is used as a bifurcation parameter. A variety of mono-, bi- and tri-stable behaviors with regular and chaotic attractors are analyzed. It is shown how random disturbances of the bifurcation parameter cause multistage stochastic transformations, noise-induced excitement, and stochastic transitions from order to chaos and reversely. These noise-induced effects are studied in terms of stochastic P- and D-bifurcations.

Jacobian-free variational method for computing connecting orbits in nonlinear dynamical systems.

One approach for describing spatiotemporal chaos is to study the unstable invariant sets embedded in the chaotic attractor of the system. While equilibria, periodic orbits, and invariant tori can be computed using existing methods, the numerical identification of heteroclinic and homoclinic connections between them remains challenging. We propose a robust matrix-free variational method for computing connecting orbits between equilibrium solutions. Instead of a common shooting-based approach, we view the identification of a connecting orbit as a minimization problem in the space of smooth curves in the state space that connect the two equilibria. In this approach, the deviation of a connecting curve from an integral curve of the vector field is penalized by a non-negative cost function. Minimization of the cost function deforms a trial curve until, at a global minimum, a connecting orbit is obtained. The method has no limitation on the dimension of the unstable manifold at the origin equilibrium and does not suffer from exponential error amplification associated with time-marching a chaotic system. Owing to adjoint-based minimization techniques, no Jacobian matrices need to be constructed. Therefore, the memory requirement scales linearly with the size of the problem, allowing the method to be applied to high-dimensional dynamical systems. The robustness of the method is demonstrated for the one-dimensional Kuramoto-Sivashinsky equation.