Kaplan-Yorke Dimension Research Articles

ON THE CHAOTIC NATURE OF A CAPUTO FRACTIONAL MATHEMATICAL MODEL OF CANCER AND ITS CROSSOVER BEHAVIORS

In this paper, a three-dimensional mathematical model of cancer, incorporating tumor cells, host normal cells, and effector immune cells, is examined. The chaotic nature of this cancer model is explored within the framework of Caputo fractional calculus. The analysis employs local stability assessment using the Matignon criteria, Lyapunov exponents (LEs) for various fractional orders of derivatives, bifurcation plots reflecting changes in the fractional derivative order, simulation outcomes of a novel chaotic attractor, Kaplan–Yorke dimension, and sensitivity to initial conditions. The results demonstrate that the Caputo fractional cancer model exhibits chaotic behavior for selected parameters. Moreover, the fractional derivative orders significantly influence the extent of chaotic behavior. Specifically, as the fractional derivative order increases from zero to one, the intensity of chaos diminishes, evidenced by a decrease in both the LE and the Kaplan–Yorke dimension. A piecewise modeling approach is applied to the cancer model, incorporating deterministic, stochastic, and power-law behaviors. Three distinct scenarios are developed within this framework, revealing several novel crossover behaviors through simulation. Notably, it is observed that when cancer dynamics initially follow a power-law behavior and subsequently transition into a stochastic process, the disease dynamics can stabilize, suggesting a positive outlook for disease control. Conversely, if the cancer dynamics begin with a deterministic process and then shift to a stochastic process, the system may enter a chaotic state, complicating disease management efforts.

Dynamic Analysis and FPGA Implementation of a New Linear Memristor-Based Hyperchaotic System with Strong Complexity

Chaotic or hyperchaotic systems have a significant role in engineering applications such as cryptography and secure communication, serving as primary signal generators. To ensure stronger complexity, memristors with sufficient nonlinearity are commonly incorporated into the system, suffering a limitation on the physical implementation. In this paper, we propose a new four-dimensional (4D) hyperchaotic system based on the linear memristor which is the most straightforward to implement physically. Through numerical studies, we initially demonstrate that the proposed system exhibits robust hyperchaotic behaviors under typical parameter conditions. Subsequently, we theoretically prove the existence of solid hyperchaos by combining the topological horseshoe theory with computer-assisted research. Finally, we present the realization of the proposed hyperchaotic system using an FPGA platform. This proposed system possesses two key properties. Firstly, this work suggests that the simplest memristor can also induce strong nonlinear behaviors, offering a new perspective for constructing memristive systems. Secondly, compared to existing systems, our system not only has the largest Kaplan-Yorke dimension, but also has clear advantages in areas related to engineering applications, such as the parameter range and signal bandwidth, indicating promising potential in engineering applications.

Creation of Three‐Scroll Hidden Conservative Lorenz‐Like Chaotic Flows

AbstractIn contrast to the various well‐known two‐scroll hidden Lorenz‐like attractors, little seems to be known about three‐scroll hidden conservative chaotic flows in the 3D Lorenz‐like system family whose Kaplan–Yorke dimension is approximately equal to . This technical note first proposes a three‐scroll chaotic system, i.e., , , , which generates three‐scroll hidden chaotic flows without equilibrium points when . Moreover, its Kaplan–Yorke dimension is . Then, for and , this system has another type of singularly degenerate heteroclinic cycle consisting of two different equilibria in the lines of semi‐hyperbolic equilibria and , i.e., and , with . With a small perturbation of , the kind of three‐scroll hidden conservative chaotic flows described above are created. Finally, five cycles are obtained when the proposed system undergoes Hopf bifurcation.

Qualitative Properties of a Physically Extended Six-Dimensional Lorenz System

In this paper, the qualitative properties of a physically extended six-dimensional Lorenz system, with additional physical terms describing rotation and density, which was proposed in [Moon et al., 2019] have been investigated. The dissipation, invariance, Lyapunov exponents, Kaplan–Yorke dimension, ultimate boundedness and global attractivity of this six-dimensional Lorenz system have been discussed in detail according to the chaotic systems theory. We find that this system exhibits chaos phenomena for a new set of parameters. It is well known that the general method for studying the bounds of a chaotic system is to construct a suitable Lyapunov-like function (or the generalized positive definite and radically unbounded Lyapunov function). However, the higher the dimension of a chaotic system, the more difficult it is to construct the Lyapunov-like function. The innovation of this paper is that we first construct the suitable Lyapunov-like function for this six-dimensional Lorenz system, and then we prove that this system is not only globally bounded for varying parameters, but it also gives a collection of global absorbing sets for this system with respect to all parameters of this system according to Lyapunov’s direct method and the optimization method. Furthermore, we obtain the rate of the trajectories going from the exterior to the global absorbing set. Some numerical simulations are presented to validate our research results. Finally, we give a direct application of the results obtained in this paper. According to the results of this paper, we can conclude that the equilibrium point [Formula: see text] of this system is globally exponentially stable.

Analysis and Applications of the Chaotic Hyperbolic Memristor Model with Fractional Order Derivative

This work presents a new five-dimensional fractional-order chaotic system that includes a feedback memristor. A Lorenz-Stenflo-based fractional-order chaotic system with five dimensions that contains feedback memristor dynamics is used to do this. This work focuses on chaotic models through the ABC fractional derivative, a concept we rigorously examine. We designed and usedsophisticated numerical algorithms to model fractional-order dynamics with the utmost precision to understand this system’s complex characteristics. These numerical approaches excel at non-integer order differentiation, which standard numerical methods struggle with. Our research uses fractional calculus to increase the system’s complexity and robustness, revealing its hyperchaotic nature. We demonstrate that the system is chaotic and stable over fractional orders using eigenvalues, Lyapunov exponents, Kaplan-Yorke dimensions, maximal exponents, phase portraits, and equilibrium points. Our conclusions are strengthened by using these numerical systems, intended for this work, and analytical tools. We improve our understanding of fractional-order chaotic systems with our research. It illuminates how to improve electrical integrations and create more sophisticated nonlinear dynamical systems. This work establishes the approach and insights for future research, guiding the development of new systems with enhanced dynamical features

Predator–Prey Interaction with Fear Effects: Stability, Bifurcation and Two-Parameter Analysis Incorporating Complex and Fractal Behavior

This study investigates the dynamics of predator–prey interactions with non-overlapping generations under the influence of fear effects, a crucial factor in ecological research. We propose a novel discrete-time model that addresses limitations of previous models by explicitly incorporating fear. Our primary question is: How does fear influence the stability of predator–prey populations and the potential for chaotic dynamics? We analyze the model to identify biologically relevant equilibria (fixed points) and determine the conditions for their stability. Bifurcation analysis reveals how changes in fear levels and predation rates can lead to population crashes (transcritical bifurcation) and complex population fluctuations (period-doubling and Neimark–Sacker bifurcations). Furthermore, we explore the potential for controlling chaotic behavior using established methods. Finally, two-parameter analysis employing Lyapunov exponents, spectrum, and Kaplan–Yorke dimension quantifies the chaotic dynamics of the proposed system across a range of fear and predation levels. Numerical simulations support the theoretical findings. This study offers valuable insights into the impact of fear on predator–prey dynamics and paves the way for further exploration of chaos control in ecological models.

Small-scale variation of atmospheric dynamics applying chaos theory, case study

Characterization and knowledge of the variability of atmospheric dynamics on a small scale in the city of Riobamba, Ecuador, are achieved through the chaos theory. Meteorological data is taken every hour during four years, including variables such as wind speed, wind direction, incident radiation, temperature, and humidity, from the ESPOCH, SAN JUAN, and QUIMIAG weather stations in the canton of Riobamba. The van Ulden and Hostlang models are used to calculate the Obukhov length, surface heat fluxes, and latent heat flux. The chaos theory is applied to study the variation of atmospheric microdynamics. The Lyapunov coefficients, Kolmogorov-Sinai entropy, and Kaplan-Yorke fractal dimension are determined. Before analysis, noise reduction is necessary due to the lack of correlation, especially in the Obukhov length. This research follows a longitudinal design and employs quantitative and explanatory methods based on data analysis, statistical-mathematical techniques, and inductive-deductive approaches. The results indicate a highly variable system, reflected in a high number of Lyapunov coefficients, fractional dimensions, and entropy variations. The microdynamic parameters exhibit hyperchaotic behavior, as indicated by the presence of more than one positive Lyapunov coefficient. The variables also demonstrate a fractional fractal dimension, highlighting the irregularity in the geometric representation of the system.

A Nonlinear Megastable System with Diamond-Shaped Oscillators

Benefiting from trigonometric and hyperbolic functions, a nonlinear megastable chaotic system is reported in this paper. Its nonlinear equations without linear terms make the system dynamics much more complex. Its coexisting attractors’ shape is diamond-like; thus, this system is said to have diamond-shaped oscillators. State space and time series plots show the existence of coexisting chaotic attractors. The autonomous version of this system was studied previously. Inspired by the former work and applying a forcing term to this system, its dynamics are studied. All forcing term parameters’ impacts are investigated alongside the initial condition-dependent behaviors to confirm the system’s megastability. The dynamical analysis utilizes one-dimensional and two-dimensional bifurcation diagrams, Lyapunov exponents, Kaplan–Yorke dimension, and attraction basin. Because of this system’s megastability, the one-dimensional bifurcation diagrams and Kaplan–Yorke dimension are plotted with three distinct initial conditions. Its analog circuit is simulated in the OrCAD environment to confirm the numerical simulations’ correctness.

A New 3D Chaotic Attractor in Gene Regulatory Network

This paper introduces a new 3D chaotic attractor in a gene regulatory network. The proposed model has eighteen parameters. Formulas for characteristic numbers of critical points for three-dimensional systems were considered. We show that the three equilibrium points of the new chaotic 3D system are unstable and deduce that the three-dimensional system exhibits chaotic behavior. The possible outcomes of this 3D model were compared with the results of the Chua circuit. The bifurcation structures of the proposed 3D system are investigated numerically, showing periodic solutions and chaotic solutions. Lyapunov exponents and Kaplan-Yorke dimension are calculated. For calculations, the Wolfram Mathematica is used.

On chaos control of nonlinear fractional Newton-Leipnik system via fractional Caputo-Fabrizio derivatives

In this work, we present a design for a Newton-Leipnik system with a fractional Caputo-Fabrizio derivative to explain its chaotic characteristics. This time-varying fractional Caputo-Fabrizio derivative approach is applied to solve the model numerically, and to check the solution’s existence and uniqueness. The existence and uniqueness of results of a fractional-order model under the Caputo-Fabrizio fractional operator have been proved by fixed point theory. As well, we achieved a stable result by applying the Ulam-Hyers concept. Chaos is controlled by linear controllers. Furthermore, the Lyapunov exponent of the system indicates that the chaos control findings are accurate. Based on weighted covariant Lyapunov vectors we construct a background covariance matrix using the Kaplan-Yorke dimension. Using a numerical example, this suggested method is illustrated for its applicability and efficiency.

Chaotic dynamics of diatomic systems in an electromagnetic field: Dynamical and topological invariants

An advanced combined quantum-dynamic and chaos-geometric method for analysis, modelling, and forecasting of the chaotic dynamics of diatomic molecules in an intense electromagnetic field is presented. The method is based on the use of the non-stationary theory of the Schrödinger equation in the approximation of the density functional and the methods of the theory of chaos and dynamic systems for the analysis of time series of polarization and other characteristics of diatomic molecules in an intense electromagnetic field. In particular, the latter includes the Gottwald-Melbourne test, the correlation integral method , fractal and multifractal formalism, average mutual information, false nearest neighbours, surrogate data algorithms, analysis on the basis of the Lyapunov's exponents, Kolmogorov entropy, nonlinear forecast models based on algorithms of optimized predicted trajectories, B-spline approximations. As an illustration, the advanced data for the dynamical and topological invariants (correlation dimension, embedding dimension, Kaplan-York dimension, Lyapunov's exponents, Kolmogorov entropy, etc.) for the diatomic ZrO molecule in a linearly polarized electromagnetic field are listed.

Analysis and Chaos Control and Synchronization of the Nonlinear Newton–Leipnik System Via Feedback Adaptive Control Techniques

This paper presents the system analysis and chaos control synchronization of the Newton–Leipnik system. Uniqueness and existence solutions (equilibrium points or fixed points) of Newton–Leipnik system have been discussed and local analysis of the system at each equilibrium points is studied. Lyapunov exponents and Kaplan–Yorke dimensions, bifurcation diagrams and poincare sections are analyzed and plotted to establish the presence of chaos in the system. Control feedback techniques are simulated in the nonlinear Newton–Leipnik system. Controlling chaos of systems to find the desired result are established. Synchronization methodologies of two identical integer-order Newton–Leipnik systems are yielded applying adaptive control technologies which have been verified through plotting the graphs of tracking the trajectories of master to slave systems, parameter identification results and synchronization of error dynamics diagrams of Newton–Leipnik systems. These findings offer valuable insights and tools applicable to various scientific and technological domains. These results highlight the originality of our study and deepen our understanding of chaos in Newton–Leipnik systems, offering practical applications and enhanced insights for researchers and scientists in chaos analysis.

Reconstruction, forecasting, and stability of chaotic dynamics from partial data.

The forecasting and computation of the stability of chaotic systems from partial observations are tasks for which traditional equation-based methods may not be suitable. In this computational paper, we propose data-driven methods to (i) infer the dynamics of unobserved (hidden) chaotic variables (full-state reconstruction); (ii) time forecast the evolution of the full state; and (iii) infer the stability properties of the full state. The tasks are performed with long short-term memory (LSTM) networks, which are trained with observations (data) limited to only part of the state: (i) the low-to-high resolution LSTM (LH-LSTM), which takes partial observations as training input, and requires access to the full system state when computing the loss; and (ii) the physics-informed LSTM (PI-LSTM), which is designed to combine partial observations with the integral formulation of the dynamical system's evolution equations. First, we derive the Jacobian of the LSTMs. Second, we analyze a chaotic partial differential equation, the Kuramoto-Sivashinsky, and the Lorenz-96 system. We show that the proposed networks can forecast the hidden variables, both time-accurately and statistically. The Lyapunov exponents and covariant Lyapunov vectors, which characterize the stability of the chaotic attractors, are correctly inferred from partial observations. Third, the PI-LSTM outperforms the LH-LSTM by successfully reconstructing the hidden chaotic dynamics when the input dimension is smaller or similar to the Kaplan-Yorke dimension of the attractor. The performance is also analyzed against noisy data. This work opens new opportunities for reconstructing the full state, inferring hidden variables, and computing the stability of chaotic systems from partial data.

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.

A non-autonomous mega-extreme multistable chaotic system

Megastable and extreme multistable systems comprise two major new branches of multistable systems. So far, they have been studied separately in various chaotic systems. Nevertheless, to the best of our knowledge, no chaotic system has so far been reported that possesses both types of multistability. This paper introduces the first three-dimensional non-autonomous chaotic system that displays megastability and extreme multistability, jointly called mega-extreme multistability. Our model shows extreme multistability for a variation of an initial condition associated with one system variable and megastability concerning another variable. The different types of coexisting attractors are characterized by the corresponding phase portraits and first return maps, as well as by constructing the appropriate bifurcation diagrams, calculating the Lyapunov spectra, the Kaplan-Yorke dimension and the connecting curves, and by determining the corresponding basins of attraction. The system is explicitly shown to be dissipative, with the dissipation being state-dependent. We demonstrate the feasibility and applicability of our model by designing and simulating an appropriate analog circuit.

A New 8D Lorenz-like Hyperchaotic System: Computer Modelling, Circuit Design and Arduino Uno Board Implementation

This paper presents the construction of Matlab-Simulink and LabView models for a novel nonlinear dynamic system of equations in an eight-dimensional (8D) phase space. The Lyapunov exponent spectrum and Kaplan-York dimension were calculated with fixed parameters of the 8D dynamical system. The presence of two positive Lyapunov exponents indicates hyperchaotic behavior. The fractional Kaplan-York dimension shows the fractal structure of strange attractors. An adaptive controller was used to stabilize the 8D chaotic system with unknown system parameters, and an active control method was derived to achieve global chaotic synchronization of two identical 8D chaotic systems with unknown system parameters. Using the results from Matlab-Simulink and LabView models, a chaotic signal generator for the 8D chaotic system was implemented in the Multisim environment. The simulation results of chaotic behavior in the Multisim environment demonstrate similar behavior compared to simulation results from Matlab-Simulink and LabView models. To visualize the new 8D chaotic system, we employed an Arduino Uno board along with eight light-emitting diodes (LEDs). Furthermore, we demonstrated the capability to simulate the new 8D chaotic system in the Proteus 8 environment using the Arduino Uno microcontroller. This advancement in the understanding and implementation of chaotic systems opens doors to numerous possibilities in the realm of secure communication and control systems.

A Novel Chaotic System with Exponential Nonlinearity and its Adaptive Self-Synchronization: From Numerical Simulations to Circuit Implementation

In this paper, a new three-dimensional chaotic system with two exponential nonlinearities is presented. The analysis of fixed points of the proposed system suggests existence of one hyperbolic index-2 spiral saddle-type fixed point. The proposed system fulfils Shilnikov criterion and nature of the chaos is found to be dissipative. Bifurcation diagrams, maximum Lyapunov exponents and Kaplan–Yorke dimension are examined through numerical simulations to investigate dynamics of proposed system. The hardware feasibility of the proposed system is illustrated through current feedback operational amplifier (CFOA)-based circuit implementation. The proposed circuit uses six CFOAs, two diodes to establish nonlinearity, eleven resistors and three capacitors. The absence of analog multiplier in the proposed circuit makes it superior to the existing counterpart in the sense that it does not require area and power-consuming active building block. To confirm the chaotic nature of the proposed circuit, LTspice simulations are done to obtain phase portraits which are found to be strange attractors and are topologically different from the shape of the existing attractors. Moreover, we have investigated the synchronization of the proposed chaotic system using adaptive control scheme and proposed CFOA-based complete circuit design of the adaptively synchronized system. Also, the effect of the tolerance of passive components and temperature on the behavior of the proposed chaotic circuit and the complete synchronization circuit has also been studied. It is found that the circuit is sensitive to the value of resistors and temperature to an extent and can work properly within their limits.

Analysis and description S-box generation for the AES algorithm-a new 3D hyperchaotic system

In this paper, a description, and analysis of a novel 3-D dimension hyperchaotic system is implemented. The proposed system oscillation is two-order autonomous and consisted of a nine-term and symmetric oscillation w.r.t x-axis. It is proved analysis by Kaplan-York dimension, waveform analysis, phase portrait, and Lyapunov exponent. This work-study stability and equilibrium point and Routh stability criteria produced that the new system has one unstable point from the type saddle-focus point. One of the characteristics of the proposed system is hyperchaotic since this system has two Lyapunov large than zero. This system is applied to generate a chaotic (S-box) based in advanced encryption standard (AES) algorithm for text encryption and gives a high level of security. In addition to the description, and analysis S-box. Therefore. the proposed algorithm is satisfied the high randomness of entropy value and passes the National Institute of Standards and Technology (NIST) parameters and another test. Mathematica and MATLAB programs simulated some results.

Coexisting attractors and multi-stability within a Lorenz model with periodic heating function

In this paper, the classical Lorenz model is under investigation, in which a periodic heating term replaces the constant one. Applying the variable heating term causes time-dependent behaviors in the Lorenz model. The time series produced by this model are chaotic; however, they have fixed point or periodic-like qualities in some time intervals. The energy dissipation and equilibrium points are examined comprehensively. This modified Lorenz system can demonstrate multiple kinds of coexisting attractors by changing its initial conditions and, thus, is a multi-stable system. Because of multi-stability, the bifurcation diagrams are plotted with three different methods, and the dynamical analysis is completed by studying the Lyapunov exponents and Kaplan-Yorke dimension diagrams. Also, the attraction basin of the modified system is investigated, which approves the appearance of coexisting attractors in this system.

On a New Three-Dimensional Chaotic System with Adaptive Control and Chaos Synchronization

A three-dimensional autonomous deterministic chaotic system having six parameters is explored within this article. The dynamical characteristics of the proposed system are investigated through eigenvalues structure, bifurcation diagrams, Kaplan–Yorke dimension, Lyapunov exponents, time response, and phase plane trajectories. For the suitable design of the system parameters, it is found that the system can exhibit periodic, period-n, or chaotic oscillations. Accordingly, the system’s dynamical behavior to the variation of its coefficients has been explored. The obtained results revealed that the proposed dynamical system does not lose its chaotic oscillations for the small fluctuations of one or more of the values of its parameters. In addition, chaos control and chaos synchronization have been studied by means of the adaptive control strategy relying on Lyapunov’s second method of stability. The numerical simulation revealed that superior chaos control and master-slave synchronization have been achieved by the applied control laws. Finally, the obtained results have been simulated via a nonlinear electronic circuit that demonstrated the feasibility of the purposed chaotic system for different engineering applications such as secure communications, cryptosystems, image encryption, and image processing.