Infinite Number Of Equilibrium Points Research Articles

Equilibrium space and a pseudo linearization of nonlinear systems

This paper attempts to extend the concept of the equilibrium point to what is called equilibrium space, which can adapt to a system in which there exists an infinite number of equilibrium points. In the context of Lyapunov’s linearization method extended for the equilibrium space, this paper proposes a pseudo linearization, from which we can derive a linear representation for a nonlinear system. The equilibrium state of this pseudo linearization and its stability are shown to be the same as that of the original nonlinear system. As an example of the applicability, the proposed pseudo linearization is applied to derive a discrete-time model for a control moment gyroscope system from a nonlinear continuous-time model. Simulation results show that the discrete-time model derived using the proposed pseudo linearization yields responses that are closer to that of the continuous-time model than the discrete-time model derived by the well-known forward-difference method and the conventional pseudo linear representation method, even with a large sampling interval.

A Simple Method for Constructing a Family of Hamiltonian Conservative Chaotic Systems

Conservative chaotic systems (CCSs) have unique advantages over dissipative chaotic systems (DCSs) in the fields of secure communication and pseudo-random number generators (PRNGs), etc. However, there are relatively fewer reports on CCSs than DCSs. To this end, this paper proposes an effective method for constructing a family of Hamiltonian conservative chaotic systems (HCCSs) by letting any three of the four sub-bodies denoted by 4D generalized Euler equations share a rotation axis. From theoretical analysis to experimental verification, one of the proposed HCCSs is studied thoroughly to demonstrate the effectiveness of this method. The example system has an infinite number of equilibrium points, which belong to either centers or saddles, resulting in hidden chaos. Besides, through the bifurcation diagram, parametric chaotic set, and Lyapunov exponent, richly dynamic behaviors related to parameters are displayed. Moreover, the 3D phase portraits verify that the Hamiltonian energy is conservative. Numerous energy-related coexisting orbits are discovered in this system, such as the coexistence of quasi-periodic orbits, chaotic orbits, and chaotic quasi-periodic orbits. Furthermore, the breadboard-based circuit is implemented to illustrate the HCCS’s physical feasibility. Finally, the PRNG based on the HCCS has excellent randomness in terms of NIST and TESTU01 test results.

A New Chaotic System with Only Nonhyperbolic Equilibrium Points: Dynamics and Its Engineering Application

In this work, we introduce a new non-Shilnikov chaotic system with an infinite number of nonhyperbolic equilibrium points. The proposed system does not have any linear term, and it is worth noting that the new system has one equilibrium point with triple zero eigenvalues at the origin. Also, the novel system has an infinite number of equilibrium points with double zero eigenvalues that are located on the z -axis. Numerical analysis of the system reveals many strong dynamics. The new system exhibits multistability and antimonotonicity. Multistability implies the coexistence of many periodic, limit cycle, and chaotic attractors under different initial values. Also, bifurcation analysis of the system shows interesting phenomena such as periodic window, period-doubling route to chaos, and inverse period-doubling bifurcations. Moreover, the complexity of the system is analyzed by computing spectral entropy. The spectral entropy distribution under different initial values is very scattered and shows that the new system has numerous multiple attractors. Finally, chaos-based encoding/decoding algorithms for secure data transmission are developed by designing a state chain diagram, which indicates the applicability of the new chaotic system.

A novel double-wing chaotic system with infinite equilibria and coexisting rotating attractors: Application to weak signal detection

By introducing a sine function, a non-autonomous multi-wing chaotic system is proposed. The system has an infinite number of equilibrium points and produces symmetrical attractors. The complex dynamical behaviors of the system are demonstrated by phase portraits, Lyapunov exponents spectrum and bifurcation diagram. The effect of driving amplitudes and initial conditions on the resulting system dynamics is then thoroughly investigated. The resulting attractors will enter different oscillatory states or have topological changes. The rotational coexisting attractors depend on the initial conditions and external amplitudes. Besides, a variety of interesting symmetrical transient behaviors and initial-offset boosting behaviors are also found. The driving amplitudes of the system affect the number of attractor wings, a weak signal detection circuit is accordingly designed to estimate the amplitude of periodic weak signals at diverse frequencies. The circuit operates near the switching threshold between the four-wing and two-wing chaotic attractor. Finally, an experimental investigation of the proposed design is performed that demonstrates the theoretical and simulation results.

A Novel Non-Autonomous Chaotic System With Infinite 2-D Lattice of Attractors and Bursting Oscillations

In this brief, a novel three-dimensional non-autonomous chaotic system with periodic excitation and trigonometric function is proposed. Interestingly, with the disturbance of the periodic excitation, the system exhibits complex dynamical behaviors, including bursting oscillations (BOs), chaotic and hyperchaotic attractor. More importantly, because of the presence of trigonometric function, the system possesses infinite number of equilibrium points, which leads to the phenomenon of extreme multistability, namely infinite coexistence attractors and BOs. Besides, a variety of dynamic analysis tools such as phase diagram (PD), transformed phase diagram (TPD), time series (TS), bifurcation diagram (BD) and Lyapunov exponents (LE) are used to comprehensively analyze these interesting dynamics. Finally, an analog circuit is designed through the use of circuit simulation software PSPICE and realized by an experimental set-up to verify these dynamics.

Fractional-Order Hidden Attractor Based on the Extended Liu System

In this paper, a new commensurate fractional-order chaotic oscillator is presented. The mathematical model with a weak feedback term, which is named hypogenetic flow, is proposed based on the Liu system. And with changing the parameters of the system, the hidden attractor can have no equilibrium points or line equilibrium. What is more interesting is that under the occasion that no equilibrium point can be obtained, the phase trajectory can converge to a minimal field under the lead of some initial conditions, similar to the fixed point. We call it the virtual equilibrium point. On the other hand, when the value of parameters can produce an infinite number of equilibrium points, the line equilibrium points are nonhyperbolic. Moreover than that, there are coexistence attractors, which can present hyperchaos, chaos, period, and virtual equilibrium point. The dynamic characteristics of the system are analyzed, and the parameter estimation is also studied. Then, an electronic circuit implementation of the system is built, which shows the feasibility of the system. At last, for the fractional system with hidden attractors, the finite-time synchronization control of the system is carried out based on the finite-time stability theory of the fractional system. And the effectiveness of the controller is verified by numerical simulation.

A novel meminductor-based chaotic oscillating circuit and its DSP realisation for generating PN sequences

Meminductor is a novel nonlinear inductor with memory. A meminductor-based chaotic oscillating circuit that has only two linear resistors, two linear capacitors and a meminductor, is designed based on a mathematical model of the flux-controlled meminductor to study its characteristics in nonlinear circuit. Through the analysis of bifurcations, dynamic map and Lyapunov exponents, it is found that the system can exhibit some complex characteristics, such as an infinite number of equilibrium points and burst chaos. Especially, bifurcation without parameters and coexisting attractors appear under a fixed set of parameter values but different initial conditions. Moreover, random characteristics of the PN sequences generated from the chaotic circuit are tested via the test suit of National Institute of Standards and Technology (NIST), and the tested randomness definitely reaches the standards of NIST. Finally, a scheme for digitally realising this oscillating circuit is provided using the digital signal processor (DSP).

Four-Wing Hidden Attractors with One Stable Equilibrium Point

Although multiwing hidden attractor chaotic systems have attracted a lot of interest, the currently reported multiwing hidden attractor chaotic systems are either with no equilibrium point or with an infinite number of equilibrium points. The multiwing hidden attractor chaotic systems with stable equilibrium points have not been reported. This paper reports a four-wing hidden attractor chaotic system, which has only one stable node-focus equilibrium point. The novel system can also generate a hidden attractor with one-wing and hidden attractors with quasi-periodic and periodic coexistence. In addition, a self-excited attractor with one-wing can be generated by adjusting the parameters of the novel system. The hidden attractors of the novel system are verified by the cross-section of attraction basins. And the hidden behavior is investigated by choosing different initial states. Moreover, the coexisting transient four-wing phenomenon of the self-excited one-wing attractor system is studied by the time domain waveforms and attraction basin. The dynamical characteristics of the novel system are studied by Lyapunov exponents spectrum, bifurcation diagram and Poincaré map. Furthermore, the novel hidden attractor system with four-wing and one-wing are implemented by electronic circuits. The hardware experiment results are consistent with the numerical simulations.

A Nonvolatile Fractional Order Memristor Model and its Complex Dynamics

It is found that the fractional order memristor model can better simulate the characteristics of memristors and that chaotic circuits based on fractional order memristors also exhibit abundant dynamic behavior. This paper proposes an active fractional order memristor model and analyzes the electrical characteristics of the memristor via Power-Off Plot and Dynamic Road Map. We find that the fractional order memristor has continually stable states and is therefore nonvolatile. We also show that the memristor can be switched from one stable state to another under the excitation of appropriate voltage pulse. The volt–ampere hysteretic curves, frequency characteristics, and active characteristics of integral order and fractional order memristors are compared and analyzed. Based on the fractional order memristor and fractional order capacitor and inductor, we construct a chaotic circuit, of which the dynamic characteristics with respect to memristor’s parameters, fractional order α, and initial values are analyzed. The chaotic circuit has an infinite number of equilibrium points with multi-stability and exhibits coexisting bifurcations and coexisting attractors. Finally, the fractional order memristor-based chaotic circuit is verified by circuit simulations and DSP experiments.

Scalable Robust Voltage Control of DC Microgrids With Uncertain Constant Power Loads

Constant power loads (CPLs) impose instability issues in DC microgrids due to their negative impedance characteristics. This paper studies the problem of voltage control design of DC microgrids with CPLs. It is assumed that the power of CPLs is uncertain and belongs to a given interval leading to an infinite number of equilibrium points of the system. We develop a polytope model for DC microgrids with uncertain CPLs. Using this model, a robust two-degree-of-freedom feedback-feedforward voltage control framework is then proposed. The voltage controller is obtained by a solution of a set of linear matrix inequalities. The voltage control design strategy for each distributed generation (DG) unit is scalable and independent of the other DGs. The effectiveness of the proposed control approach is evaluated through simulation studies in MATLAB/SimPowerSystems toolbox.

A New Family of Chaotic Systems with Different Closed Curve Equilibrium

Chaotic systems with hidden attractors, infinite number of equilibrium points and different closed curve equilibrium have received much attention in the past six years. In this work, we introduce a new family of chaotic systems with different closed curve equilibrium. Using the methods of equilibrium points, phase portraits, maximal Lyapunov exponents, Kaplan–Yorke dimension, and eigenvalues, we analyze the dynamical properties of the proposed systems and extend the general knowledge of such systems.

Robe’s restricted three-body problem when one of the primaries is a finite straight segment

In this paper we study the existence and linear stability of the equilibrium points in the Robe’s restricted three-body problem when one of the primaries is a finite straight segment. We observe that the center of the first primary is always an equilibrium point for all values of the density parameter k, mass parameter μ and length parameter l. There is an additional equilibrium point lying on the line joining the center of the first primary and the second primary, provided k>1+l2. When k+μ=1+l2(1−μ), there are infinite number of equilibrium points lying on a circle with radius 1−23l2 and center as the second primary, provided they lie within the spherical shell. When k<0 and k+μ+2μl2>0, there are two more equilibrium points lying in the xz-plane forming triangles with the center of the shell and the second primary. We analyze the linear stability of the equilibrium points.

Stability analysis and variable speed limit control of a traffic flow model

The cell transmission traffic flow model (CTM) has attracted considerable interest in the field of transportation due to its simplicity as well as the ability to capture most of the macroscopic traffic flow characteristics. The stability properties of the CTM under different demand and capacity constraints are not always obvious. In addition, the impact of microscopic phenomena such as forced lane changes at bottlenecks leading to capacity drop is not captured by the CTM. In this paper, we start with a single section and modify the CTM to account for capacity drop. We analyze the stability properties of the CTM under all possible demand and capacity constraints as well as all possible initial density conditions. The analysis is used to motivate the design of variable speed limit (VSL) control to overcome capacity drop and achieve the maximum possible flow under all feasible traffic situations. The results are extended to multiple sections, where the stability properties of the open-loop system are analyzed and a VSL control scheme is designed and shown to achieve the objective of maximizing the traffic flow under different demand and capacity constraints. Unlike the open loop system where an infinite number of equilibrium points exist under certain demand levels, the proposed nonlinear VSL scheme guarantees exponential convergence to a unique equilibrium point that corresponds to maximum possible flow and speed under all possible demand levels and capacity constraints.

Hyperchaotic Memcapacitor Oscillator with Infinite Equilibria and Coexisting Attractors

A newly introduced charge-controlled memcapacitor-based hyperchaotic oscillator with coexisting chaotic attractors is investigated. Dynamic analysis of the oscillator shows that it has infinite number of equilibrium points and shows multistability. Its multistability analysis in the parameter space shows the existence of chaotic and hyperchaotic attractors. Fractional-order analysis of the hyperchaotic oscillator shows that the hyperchaos remains in the fractional order too. Field programmable gate arrays are used to realize the proposed oscillator.

СКРЫТЫЕ АТТРАКТОРЫ НЕКОТОРЫХ МУЛЬТИСТАБИЛЬНЫХ СИСТЕМ С БЕСКОНЕЧНЫМ ЧИСЛОМ СОСТОЯНИЙ РАВНОВЕСИЯ

It is well known that mathematically simple systems of nonlinear differential equations can exhibit chaotic behavior. Detection of attractors of chaotic systems is an important problem of nonlinear dynamics. Results of recent researches have made it possible to introduce the following classification of periodic and chaotic attractors depending on the presence of neighborhood of equilibrium into their basin of attraction – self-excited and hidden attractors. The presence of hidden attractors in dynamical systems has received considerable attention to both theoretical and applied research of this phenomenon. Revealing of hidden attractors in real engineering systems is extremely important, because it allows predicting the unexpected and potentially dangerous system response to perturbations in its structure. In the past three years after discovering by S. Jafari and J. C. Sprott chaotic system with a line and a plane of equilibrium with hidden attractors there has been much attention to systems with uncountable or infinite equilibria. In this paper it is offered new models of control systems with an infinite number of equilibrium possessing hidden chaotic attractors: a piecewise-linear system with a locally stable segment of equilibrium and a system with periodic nonlinearity and infinite number of equilibrium points. The original analytical-numerical method developed by the author is applied to search hidden attractors in investigated systems.

Dynamics and circuit of a chaotic system with a curve of equilibrium points

ABSTRACTAlthough chaotic systems have been intensively studied since the 1960s, new systems with mysterious features are still of interest. A novel chaotic system including hyperbolic functions is proposed in this work. Especially, the system has an infinite number of equilibrium points. Dynamics of the system are investigated by using non-linear tools such as phase portrait, bifurcation diagram, and Lyapunov exponent. It is interesting that the system can display coexisting chaotic attractors. An electronic circuit for realising the chaotic system has been implemented. Experimental results show a good agreement with theoretical ones.

Different Families of Hidden Attractors in a New Chaotic System with Variable Equilibrium

A new chaotic system having variable equilibrium is introduced in this paper. The presence of an infinite number of equilibrium points, a stable equilibrium, and no-equilibrium is observed in the system. Interestingly, this system is classified as a rare system with hidden attractors from the view point of computation. Complex dynamical behavior and a circuital implementation of the new system have been investigated in our work.

A chaotic system with an infinite number of equilibrium points located on a line and on a hyperbola and its fractional-order form

A three-dimensional autonomous chaotic system with an infinite number of equilibrium points located on a line and a hyperbola is proposed in this paper. To analyze the dynamical behaviors of the proposed system, mathematical tools such as Routh-Hurwitz criteria, Lyapunov exponents and bifurcation diagram are exploited. For a suitable choice of the parameters, the proposed system can generate periodic oscillations and chaotic attractors of different shapes such as bistable and monostable chaotic attractors. In addition, an electronic circuit is designed and implemented to verify the feasibility of the proposed system. A good qualitative agreement is shown between the numerical simulations and the Orcard-PSpice results. Moreover, the fractional-order form of the proposed system is studied using analog and numerical simulations. It is found that chaos, periodic oscillations and periodic spiking exist in this proposed system with order less than three. Then an electronic circuit is designed for the commensurate fractional order α = 0.98, from which we can observe that a chaotic attractor exists in the fractional-order form of the proposed system. Finally, the problem of drive-response generalized projective synchronization of the fractional-order form of the chaotic proposed autonomous system is considered.

Dynamics, FPGA realization and application of a chaotic system with an infinite number of equilibrium points

Recently, systems with infinite equilibria have attracted interest because they can be considered as systems with hidden attractors. In this work, we introduce a new elegant system with an open curve of equilibrium points. It has just one parameter (a) and an exponential function. We show the dynamics of such a new chaotic oscillator by computing the Lyapunov exponents’ spectrum and bifurcation diagram, and it is implemented by using a field-programmable gate array (FPGA). The exponential function is approached by power series and then implemented with adders and multipliers within the FPGA. Experimental results are provided for the attractor as well as for the synchronization of two chaotic oscillators to transmit an image, thus demonstrating the usefulness of the new oscillator in a chaotic secure communication system.

Stability Analysis and Control of Nonholonomic Systems with Potential Fields

This paper presents a theoretical analysis on the stability of nonholonomic systems when a potential field method is applied to them, and shows a numerical example for a nonholonomic underwater vehicle among obstacles. Nonholonomic systems with a potential function have an infinite number of equilibrium points, because the motion of the systems cannot always be generated exactly along the gradient vector of the potential function. By utilizing the component of the input that does not increase or decrease the potential function, the equilibrium points other than the critical points of the function can be destabilized, if the controllability of the systems is satisfied with the first-order Lie brackets of input vector fields. A time-invariant controller is proposed based on the theoretical analysis on the stability of equilibrium points, and applied to an underwater vehicle among obstacles. When the potential function has saddles as its critical points, the potential function is modified to be time-varying near the saddles in order to prevent the system from being trapped in the saddles. Numerical simulation results demonstrate that the underwater vehicle with the proposed controller converges to the desired point without collision with obstacles.