Infinite Number Of Equilibria Research Articles

A Novel Memristor-Based Multi-Vortex Hyper-Chaotic Circuit Design and its Application in Image Encryption

This paper proposes a new four-dimensional hyper-chaotic system capable of generating multi-wing chaotic attractors by introducing active magnetron memristors, multi-segmented square functions and trigonometric functions. The dynamical properties of this new hyper-chaotic system, such as equilibrium point, dissipation, Lyapunov exponential spectrum, bifurcation diagram and Poincaré cross-section and attraction basin, are analyzed theoretically and simulated numerically, and the complexity of this system with different parameters is analyzed. It is observed that this hyper-chaotic system has periodic, chaotic and hyper-chaotic variations with an infinite number of equilibria and coexisting attractors under different parameter conditions. The circuit simulation was performed using Multisim and the results obtained were consistent with the numerical analysis of the dynamics, and the chaotic circuit system is designed by FPGA to verify the realizability of the system. Finally, an image encryption algorithm is designed in conjunction with the DNA algorithm to enable a new system chaotic sequence for image encryption. The results show that the hyper-chaotic system has rich dynamical behavior and has high-security performance when applied to image encryption with strong chaotic key and plaintext sensitivity and large key space in image encryption.

Convergence and Equilibria Analysis of a Networked Bivirus Epidemic Model

This paper studies a networked bivirus model, in which two competing viruses spread across a network of interconnected populations; each node represents a population with a large number of individuals. The viruses may spread through possibly different network structures, and an individual cannot be simultaneously infected with both viruses. Focusing on convergence and equilibria analysis, a number of new results are provided. First, we show that for networks with generic system parameters, there exist a finite number of equilibria. Exploiting monotone systems theory, we further prove that for bivirus networks with generic system parameters, convergence to an equilibrium occurs for all initial conditions, except possibly for a set of measure zero. Given the network structure of one virus, a method is presented to construct an infinite family of network structures for the other virus that results in an infinite number of equilibria in which both viruses coexist. Necessary and sufficient conditions are derived for the local stability/instability of boundary equilibria, in which one virus is present and the other is extinct. A sufficient condition for a boundary equilibrium to be almost globally stable is presented. Then, we show how to use monotone systems theory to generate conclusions on the ordering of stable and unstable equilibria, and in some instances identify the number of equilibria via rapid simulation testing. Last, we provide an analytical method for computing equilibria in networks with only two nodes, and show that it is possible for a bivirus network to have an unstable coexistence equilibrium and two locally stable boundary equilibria.

A New Fractional-Order Map with Infinite Number of Equilibria and Its Encryption Application

The study of the chaotic dynamics in fractional-order discrete-time systems has received great attention over the last years. Some efforts have been also devoted to analyze fractional maps with special features. This paper makes a contribution to the topic by introducing a new fractional map that is characterized by both particular dynamic behaviors and specific properties related to the system equilibria. In particular, the conceived one dimensional map is algebraically simpler than all the proposed fractional maps in the literature. Using numerical simulation, we investigate the dynamic and complexity of the fractional map. The results indicate that the new one-dimensional fractional map displays various types of coexisting attractors. The approximate entropy is used to observe the changes in the sequence sequence complexity when the fractional order and system parameter. Finally, the fractional map is applied to the problem of encrypting electrophysiological signals. For the encryption process, random numbers were generated using the values of the fractional map. Some statistical tests are given to show the performance of the encryption.

Sensitivity, Equilibria, and Lyapunov Stability Analysis in Droop’s Nonlinear Differential Equation System for Batch Operation Mode of Microalgae Culture Systems

Microalgae-based biomass has been extensively studied because of its potential to produce several important biochemicals, such as lipids, proteins, carbohydrates, and pigments, for the manufacturing of value-added products, such as vitamins, bioactive compounds, and antioxidants, as well as for its applications in carbon dioxide sequestration, amongst others. There is also increasing interest in microalgae as renewable feedstock for biofuel production, inspiring a new focus on future biorefineries. This paper is dedicated to an in-depth analysis of the equilibria, stability, and sensitivity of a microalgal growth model developed by Droop (1974) for nutrient-limited batch cultivation. Two equilibrium points were found: the long-term biomass production equilibrium was found to be stable, whereas the equilibrium in the absence of biomass was found to be unstable. Simulations of estimated parameters and initial conditions using literature data were performed to relate the found results to a physical context. In conclusion, an examination of the found equilibria showed that the system does not have isolated fixed points but rather has an infinite number of equilibria, depending on the values of the minimal cell quota and initial conditions of the state variables of the model. The numerical solutions of the sensitivity functions indicate that the model outputs were more sensitive, in particular, to variations in the parameters of the half saturation constant and minimal cell quota than to variations in the maximum inorganic nutrient absorption rate and maximum growth rate.

A 3D chaotic system with piece-wise lines shape non-hyperbolic equilibria and its predefined-time control

In this paper, a novel 3D chaotic system with an infinite number of equilibria is proposed and its predefined-time control is studied. The system has non-hyperbolic equilibria with piece-wise shape, which is special and rarely mentioned before. Through 0-1 test, Lyapunov exponent, bifurcation diagram and complexity analysis, the system is deeply investigated. A reverse bubble (Feigenbaum remerging tree) is found in the system, which proves the anti-monotonicity. Furthermore, the circuit of the system is designed and the real experiment is carried out to verify its dynamic characteristics. Finally, according to the theory of predefined-time stability, a predefined time controller is designed for the system. By adding only one controller to the system, the objective of stabilizing the system within a predefined time can be achieved successfully, and simulation analysis shows good performance of the controller.

A New Memristive Chaotic System with a Plane and Two Lines of Equilibria

A new 4D memristive chaotic system with an infinite number of equilibria is proposed via exhaustive computer search. Interestingly, such a new memristive system has a plane of equilibria and two other lines of equilibria. Lyapunov exponent and bifurcation analysis show that this system has chaotic solutions with coexisting attractors. The basins of attraction of the coexisting attractors show chaos, stable fixed-points, and unbounded solutions. Furthermore, the 2D parameter space of the system is explored to find the optimum values of the parameters using the ALO (Ant Lion Optimizer) optimization algorithm.

Hidden attractors from the switching linear systems

Recently, chaotic behavior has been studied in dynamical systems that generates hidden attractors. Most of these systems have quadratic nonlinearities. This paper introduces a new methodology to develop a family of three-dimensional hidden attractors from the switching of linear systems. This methodology allows to obtain strange attractors with only one stable equilibrium, attractors with an infinite number of equilibria or attractors without equilibrium. The main matrix and the augmented matrix of every linear system are considered in Rouché-Frobenius theorem to analyze the equilibrium of the switching systems. Also, a systematic search assisted by a computer is used to find the chaotic behavior. Basic chaotic properties of the attractors are verified by the Lyapunov exponents.

A chaotic map with infinite number of equilibria in a bounded domain

A chaotic map with infinite number of equilibria in a bounded domain

Memristor-based novel complex-valued chaotic system and its projective synchronisation using nonlinear active control technique

In this paper, a flux controlled memristor-based novel complex-valued chaotic system and its projective synchronisation is investigated. The proposed complex-valued chaotic system has line and plane of equilibria, i.e. an infinite number of equilibria. Different qualitative and quantitative tools such as time series, phase plane, Poincare section, Lyapunov exponents, Lyapunov spectrum, and Lyapunov dimension are used to evidence the chaotic behaviour of the proposed complex-valued system. Further, the projective synchronisation between the proposed complex-valued chaotic systems is achieved using nonlinear active control. Active control laws are designed, by using sum of the relevant variables of the proposed complex-valued chaotic systems, to ensure the convergence of error dynamics. The required global asymptotic stability condition is derived using Lyapunov stability theory. Simulation is done in MATLAB environment to verify the theoretical approach. Simulation results reveal that the objectives of the paper are achieved successfully.

A Chaotic System with Infinite Equilibria and Its S-Box Constructing Application

Systems with many equilibrium points have attracted considerable interest recently. A chaotic system with a line equilibrium has been studied in this work. The system has infinite equilibria and exhibits coexisting chaotic attractors. The system with an infinite number of equilibria has been realized by an electronic circuit, which confirms the feasibility of the system. Based on such a system, we have developed a new S-Box generation algorithm. With the developed algorithm, two new S-Boxes are produced. Performance tests of S-Boxes are performed. The tests have shown that proposed S-Boxes have good performance results.

Memcapacitor model and its application in chaotic oscillator with memristor.

Memristors and memcapacitors are two new nonlinear elements with memory. In this paper, we present a Hewlett-Packard memristor model and a charge-controlled memcapacitor model and design a new chaotic oscillator based on the two models for exploring the characteristics of memristors and memcapacitors in nonlinear circuits. Furthermore, many basic dynamical behaviors of the oscillator, including equilibrium sets, Lyapunov exponent spectrums, and bifurcations with various circuit parameters, are investigated theoretically and numerically. Our analysis results show that the proposed oscillator possesses complex dynamics such as an infinite number of equilibria, coexistence oscillation, and multi-stability. Finally, a discrete model of the chaotic oscillator is given and the main statistical properties of this oscillator are verified via Digital Signal Processing chip experiments and National Institute of Standards and Technology tests.

Renewable Resource Use and Nonseparable Amenity Benefits

We incorporate amenity benefits into an overlapping generations model with a renewable resource as a factor of production, source of amenity benefits and store of value. Unlike the conventional renewable resource problems studied under the assumption of additive consumption and amenity benefits, we let amenity benefits affect the utility of consumers in a nonseparable fashion. We examine the role that weights given to consumption and amenities have for harvesting and the resource stock. We characterize dynamics and stability of steady state equilibria with a logistic resource growth function. We demonstrate in parametric and numerical models that the weights given to consumption and amenities in the utility function matter substantially for the steady state equilibrium stock and its stability and dynamics. Both conventional saddle point equilibria and indeterminacy with infinite number of equilibria and saddle-node bifurcation is possible depending on the weights given to consumption and amenities. In addition, we show that for each inefficient equilibrium stock, there is a unique subsidy rate that can move the economy from an inefficient equilibrium to an efficient one. The presence of indeterminacy provides a challenge to resource policies, because the system becomes unpredictable. Therefore, expectations and market psychology may play an important role in resource utilization and provision of amenities.

A Chaotic System with Different Shapes of Equilibria

Although many chaotic systems have been introduced in the literature, a few of them possess uncountably infinite equilibrium points. The aim of our short work is to widen the current knowledge of the chaotic systems with an infinite number of equilibria. A three-dimensional system with special properties, for example, exhibiting chaotic attractor with circular equilibrium, chaotic attractor with ellipse equilibrium, chaotic attractor with square-shaped equilibrium, and chaotic attractor with rectangle-shaped equilibrium, is proposed.

Morse Decompositions with Infinite Components for Multivalued Semiflows

In this paper we study the theory of Morse decompositions with an infinite number of components in the multivalued framework, proving that for a disjoint infinite family of weakly invariant sets (being all isolated but one) a Lyapunov function ordering them exists if and only if the multivalued semiflow is dynamically gradient. Moreover, these properties are equivalent to the existence of a Morse decomposition. This theorem is applied to a reaction-diffusion inclusion with an infinite number of equilibria.

Constant Returns to Scale: Can the Neoclassical Economy Exist?

This note argues that the competitive paradigm of neoclassical economics breaks down in the presence of constant returns to scale (CRS). With CRS, all goods can be produced at identical unit costs by all economic agents, making self-production a feasible alternative to market production. In the event, an infinite number of equilibria become possible with a mix of markets and self-production. If labor is the only factor of production, self-production is the only option and the market economy ceases to exist. All too often the competitive paradigm of neoclassical economics has been faulted for its unrealistic assumptions, but rarely for logical inconsistency. These critics have also developed several new variants of economics based on more realistic assumptions. 2 Nevertheless, these critiques have failed to dislodge the competitive paradigm from its preeminent position in economics. Kenneth Arrow (1994: 451) has argued that this is because the competitive paradigm “is still the only coherent account of the entire economy.” Others point to the ideological function of neoclassical economics: in particular, its defense of the capitalism of unfettered markets. 3 This note presents a different critique: it shows that the competitive paradigm is inconsistent with constant returns to scale (CRS). At least since Leon Walras, economists have constructed mathematical models to establish the exact conditions under which a market economy – with fixed resources, tastes and technology – can attain an equilibrium that is also Pareto-efficient. CRS – or the more general assumption of non-increasing returns to scale – has been a cornerstone of all these models. Allyn Young (1928) and later Nicholas Kaldor (1967 and 1972) have shown that equilibrium economics becomes irrelevant in the presence of increasing returns to scale. We argue that neoclassical economics is in trouble even with CRS: and this is a problem that has gone unnoticed. Our concern is not with the existence – or uniqueness and stability – of solutions to the system of equations that define the neoclassical economy. Instead, we ask a more basic question. Can a market economy exist in the presence of CRS?

Equilibrium Bidding Strategy for GSP Keyword Auctions

The generalized second-price (GSP) mechanism is the most widely-used auction format in sponsored search markets. However, figuring out how to bid on GSP auctions presents major theoretical and computational challenges due to the complex nature of the auction format and the infinite number of equilibria. Our study characterizes various equilibrium bidding behaviors in GSP auctions. We develop an algorithm to identify all pure-strategy Nash equilibria and discuss their distribution in the pure-strategy space. This equilibrium distribution can help advertisers formulate bidding strategies, and help search engines calculate their expected revenues.

Indeterminacy of Competitive Equilibrium in Incomplete Market Structures with Financial Assets as an Extreme Form of Market Coordination Breakdown

Market coordination breakdowns can be formulated in terms of Incomplete Market Structures, whatever is the assumption made about market power (monopoly or competition). When the assumption of competition is made, and financial assets generate an incomplete market structure, there are, generically, an infinite number of equilibria. It is shown that after embedding the model of incomplete markets in a more general model based on the absence of double coincidence of wants, a game theoretic concept of equilibrium, like the Core, allows us to prove an equivalence theorem, i.e. to show that indeterminacy is caused by the fact that the returns of financial assets have no direct utility.

Kinetic Analysis of Ultrasonic Relaxation Spectra Caused by H‐Bond Polymerization The Two State Model and the Multistep Model

AbstractA kinetic analysis based on two different types of kinetic models has been applied to the previously published ultrasonic relaxation spectra of N‐methyl acetamide dissolved in carbon tetrachloride. The first model used is a two state model which describes the kinetics of the hydrogen bond association in terms of two states, a bonded state and a non bonded state. The other model used consists of an infinite number of elementary reaction steps in which hydrogen bonds are formed. Both models are treated in such a way that they agree with the fact that the equilibrium condition of the system is described by an infinite number of equilibria between monomers and polymers. The accuracy with which the two different models describe the measured relaxation is judged by comparing the concentration dependence of the relaxation parameters predicted from the two models with the actually measured concentration dependence of the relaxation parameters. The concentration dependence of the relaxation time predicted from both models corresponds to the one measured. The concentration dependence observed for the relaxation strength, however, can be explained only in terms of the multistep model. Finally the possibility of fitting the experimental data to the theoretical expression of the relaxation of the multistep model is discussed.