Hidden hyperchaos and electronic circuit application in a 5D self-exciting homopolar disc dynamo

Analysis and circuit implementation of a non-equilibrium fractional-order chaotic system with hidden multistability and special offset-boosting

In order to obtain a system of higher complexity, a new fractional-order chaotic system is constructed based on the Sprott system. It is noteworthy that the system has no equilibrium point yet exhibits chaotic properties and has rich dynamical behavior. Its basic properties are analyzed by Lyapunov exponents, phase diagrams, and smaller alignment index tests. The change of its state is observed by changing parameters and order, during which the new system is found to have intermittent chaos phenomena. Surprisingly, the new proposed system has a special offset-boosting phenomenon, where only a boosting-controller makes the system undergo a multi-directional offset, and the shape of the generated hidden attractor changes. In addition, changing the initial value brings kinds of coexisting attractors in the system, which proves the existence of multistability. Because the new system is very sensitive to the initial value, the complexity of the new system is calculated based on the complexity algorithm, and the initial value with higher complexity is gained by contrast. Finally, the field programmable gate array is used to implement the actual circuit of the new system to verify its feasibility. This system provides an example for the study of fractional-order chaotic systems and a complex system for fractional-order chaotic applications.

Analytical and numerical investigation of the Hindmarsh-Rose model neuronal activity

In this work, a set of nonlinear equations capable of describing the transit of the membrane potential's spiking-bursting process which is shown in experiments with a single neuron was taken into consideration. It is well known that this system, which is built on dynamical dimensionless variables, can reproduce chaos. We arrived at the chaotic number after first deriving the equilibrium point. We added different nonlocal operators to the classical model's foundation. We gave some helpful existence and uniqueness requirements for each scenario using well-known theorems like Lipchitz and linear growth. Before using the numerical solution on the model, we analyzed a general Cauchy issue for several situations, solved it numerically and then demonstrated the numerical solution's convergence. The results of numerical simulations are given.

Dynamics, Periodic Orbit Analysis, and Circuit Implementation of a New Chaotic System with Hidden Attractor

Hidden attractors are associated with multistability phenomena, which have considerable application prospects in engineering. By modifying a simple three-dimensional continuous quadratic dynamical system, this paper reports a new autonomous chaotic system with two stable node-foci that can generate double-wing hidden chaotic attractors. We discuss the rich dynamics of the proposed system, which have some interesting characteristics for different parameters and initial conditions, through the use of dynamic analysis tools such as the phase portrait, Lyapunov exponent spectrum, and bifurcation diagram. The topological classification of the periodic orbits of the system is investigated by a recently devised variational method. Symbolic dynamics of four and six letters are successfully established under two sets of system parameters, including hidden and self-excited chaotic attractors. The system is implemented by a corresponding analog electronic circuit to verify its realizability.

Taming Hyperchaos with Exact Spectral Derivative Discretization Finite Difference Discretization of a Conformable Fractional Derivative Financial System with Market Confidence and Ethics Risk

Four discrete models, using the exact spectral derivative discretization finite difference (ESDDFD) method, are proposed for a chaotic five-dimensional, conformable fractional derivative financial system incorporating ethics and market confidence. Since the system considered was recently studied using the conformable Euler finite difference (CEFD) method and found to be hyperchaotic, and the CEFD method was recently shown to be valid only at fractional index α=1, the source of the hyperchaos is in question. Through numerical experiments, illustration is presented that the hyperchaos previously detected is, in part, an artifact of the CEFD method, as it is absent from the ESDDFD models.

Various bifurcations in the development of stem cells

Cell development from an undifferentiated stem cell to a differentiated one is essential in forming an organism. In this paper, various bifurcations of a stem cell during this process are studied using a model based on Furusawa and Kaneko's hypothesis. Furusawa and Kaneko's hypothesis tells that the gene expression of stem cells is chaotic. By developing to a differentiated cell, the gene expression in more order, which is the cause of losing pluripotency. In this model, the chaotic dynamics of gene expression in the stem cells become ordered during the developments. Various patterns and bifurcation points can be seen during development. The bifurcation points and their predictions during the process of cell development are studied in this paper. Some well-known critical slowing down indicators are used to show the variations of slowness during the cell's development and predict the bifurcation points. It is vital since the unexpected changes of the state can cause a disaster. All of the indicators have a proper trend by approaching the bifurcation points and faring away.

Connecting curve: A new tool for locating hidden attractors

Attractors in nonlinear dynamical systems can be categorized as self-excited attractors and hidden attractors. In contrast to self-excited attractors, which can be located by the standard numerical computational method, hidden attractors are hard to detect due to the fact that its basin of attraction is away from the proximity to equilibrium. In multistable systems, many attractors, including self-excited and hidden ones, co-exist, which makes locating each different oscillation more difficult. Hidden attractors are frequently connected to rare or abnormal oscillations in the system and often lead to unpredicted behaviors in many engineering applications, and, thus, the research in locating such attractors is considerably significant. Previous work has proposed several methods for locating hidden attractors but these methods all have their limitations. For example, one of the methods suggests that perpetual points are useful in locating hidden and co-existing attractors, while an in-depth examination suggests that they are insufficient in finding hidden attractors. In this study, we propose that the method of connecting curves, which is a collection of points of analytical inflection including both perpetual points and fixed points, is more reliable to search for hidden attractors. We analyze several dynamical systems using the connecting curve, and the results demonstrate that it can be used to locate hidden and co-existing oscillations.

Fractional-Order Analysis of Modified Chua’s Circuit System with the Smooth Degree of 3 and Its Microcontroller-Based Implementation with Analog Circuit Design

In the paper, we futher consider a fractional-order system from a modified Chua’s circuit system with the smooth degree of 3 proposed by Fu et al. Bifurcation analysis, multistability and coexisting attractors in the the fractional-order modified Chua’s circuit are studied. In addition, microcontroller-based circuit was implemented in real digital engineering applications by using the fractional-order Chua’s circuit with the piecewise-smooth continuous system.

Non periodic oscillations, bistability, coexistence of chaos and hyperchaos in the simplest resistorless Op-Amp based Colpitts oscillator

In the framework of a project on simple circuits with unexpected high degrees of freedom, we report an autonomous microwave oscillator made of a CLC linear resonator of Colpitts type and a single general purpose operational amplifier (Op-Amp). The resonator is in a parallel coupling with the Op-Amp to build the necessary feedback loop of the oscillator. Unlike the general topology of Op-Amp-based oscillators found in the literature including almost always the presence of a negative resistance to justify the nonlinear oscillatory behavior of such circuits, our zero resistor circuit exhibits chaotic and hyperchaotic signals in GHz frequency domain, as well as many other features of complex dynamic systems, including bistability. This simplest form of Colpitts oscillator is adequate to be used as didactic model for the study of complex systems at undergraduate level. Analog and experimental results are proposed.

Differential Galois integrability obstructions for nonlinear three-dimensional differential systems

In this short communication, we deal with an integrability analysis of nonlinear three-dimensional differential systems. Right-hand sides of these systems are linear in one variable, which enables one to find explicitly a particular solution and to calculate variational equations along this solution. The conditions for the complete integrability with two functionally independent rational first integrals for B-integrability and the partial integrability are obtained from an analysis of properties of the differential Galois group of variational equations. They have a very simple form of numbers, which is necessary to check whether they are appropriate integers. An application of the obtained conditions to some exemplary nonlinear three-dimensional differential systems is shown.

Didactic model of a simple driven microwave resonant T-L circuit for chaos, multistability and antimonotonicity

A simple driven bipolar junction transistor (BJT) based two-component circuit is presented, to be used as didactic tool by Lecturers, seeking to introduce some elements of complex dynamics to undergraduate and graduate students, using familiar electronic components to avoid the traditional black-box consideration of active elements. Although the effect of the base-emitter (BE) junction is practically suppressed in the model, chaotic phenomena are detected in the circuit at high frequencies (HF), due to both the reactant behavior of the second component, a coil, and to the birth of parasitic capacitances as well as to the effect of the weak nonlinearity from the base-collector (BC) junction of the BJT, which is otherwise always neglected to the favor of the predominant but now suppressed base-emitter one. The behavior of the circuit is analyzed in terms of stability, phase space, time series and bifurcation diagrams, Lyapunov exponents, as well as frequency spectra and Poincaré map section. We find that a limit cycle attractor widens to chaotic attractors through the splitting and the inverse splitting of periods known as antimonotonicity. Coexisting bifurcations confirm the existence of multi-stability behaviors, marked by the simultaneous apparition of different attractors (periodic and chaotic ones) for the same values of system parameters and different initial conditions. This contribution provides an enriching complement in the dynamics of simple chaotic circuits functioning at high frequencies. Experimental lab results are completed with PSpice simulations and theoretical ones.

On the dynamics of a simplified canonical Chua's oscillator with smooth hyperbolic sine nonlinearity: Hyperchaos, multistability and multistability control

A simplified hyperchaotic canonical Chua's oscillator (referred as SHCCO hereafter) made of only seven terms and one nonlinear function of type hyperbolic sine is analyzed. The system is found to be self-excited, and bifurcation tools associated with the spectrum of Lyapunov exponents reveal the rich dynamical behaviors of the system including hyperchaos, torus, period-doubling route to chaos, and hysteresis when turning the system control parameters. Wide ranges of hyperchaotic dynamics are highlighted in various two-parameter spaces based on two-parameter Lyapunov diagrams. The analysis of the hysteretic window using a basin of attraction as argument reveals that the SHCCO exhibits three coexisting attractors. Laboratory measurements further confirm the performed numerical investigations and henceforth validate the mathematical model. Of most/particular interest, multistability observed in the SHCCO is further controlled based on a linear augmentation scheme. Numerical results show the effectiveness of the control strategy through annihilation of the asymmetric pair of coexisting attractors. For higher values of the coupling strength, only a unique symmetric periodic attractor survives.

Spatio-temporal numerical modeling of reaction-diffusion measles epidemic system

In this work, we investigate the numerical solution of the susceptible exposed infected and recovered measles epidemic model. We also evaluate the numerical stability and the bifurcation value of the transmission parameter from susceptibility to a disease of the proposed epidemic model. The proposed method is a chaos free finite difference scheme, which also preserves the positivity of the solution of the given epidemic model.

Multi-scroll hidden attractors with two stable equilibrium points

Multiscroll hidden attractors have attracted extensive research interest in recent years. However, the previously reported multiscroll hidden attractors belong to only one category of hidden attractors, namely, the hidden attractors without equilibrium points. Up to now, multiscroll hidden attractors with stable equilibrium points have not been reported. This paper proposes a multiscroll chaotic system with two equilibrium points. The number of scrolls can be increased by adding breakpoints of a nonlinear function. Moreover, the two equilibrium points are stable node-foci equilibrium points. According to the classification of hidden attractors, the multiscroll attractors generated by a novel system are the hidden attractors with stable equilibrium points. The dynamical characteristics of the novel system are studied using the spectrum of Lyapunov exponents, a bifurcation diagram, and a Poincaré map. Furthermore, the novel system is implemented by electronic circuits. The hardware experiment results are consistent with the numerical simulations.

A Novel Autonomous 5-D Hyperjerk RC Circuit with Hyperbolic Sine Function

A novel autonomous 5-D hyperjerk RC circuit with hyperbolic sine function is proposed in this paper. Compared to some existing 5-D systems like the 5-D Sprott B system, the 5-D Lorentz, and the Lorentz-like systems, the new system is the simplest 5-D system with complex dynamics reported to date. Its simplicity mainly relies on its nonlinear part which is synthetized using only two semiconductor diodes. The system displays only one equilibrium point and can exhibit both periodic and chaotic dynamical behavior. The complex dynamics of the system is investigated by means of bifurcation analysis. In particular, the striking phenomenon of multistability is revealed showing up to seven coexisting attractors in phase space depending solely on the system's initial state. To the best of author's knowledge, this rich dynamics has not yet been revealed in any 5-D dynamical system in general or particularly in any hyperjerk system. Pspice circuit simulations are performed to verify theoretical/numerical analysis.

A New Chaotic Flow with Hidden Attractor: The First Hyperjerk System with No Equilibrium

Discovering unknown aspects of non-equilibrium systems with hidden strange attractors is an attractive research topic. A novel quadratic hyperjerk system is introduced in this paper. It is noteworthy that this non-equilibrium system can generate hidden chaotic attractors. The essential properties of such systems are investigated by means of equilibrium points, phase portrait, bifurcation diagram, and Lyapunov exponents. In addition, a fractional-order differential equation of this new system is presented. Moreover, an electronic circuit is also designed and implemented to verify the feasibility of the theoretical model.