Bifurcation Graph Research Articles

Scaling Analysis at Transition of Chaos Driven by Euler’s Numerical Algorithm

Chaos is a nonlinear phenomenon that reveals itself everywhere in nature and in many fields of science. It has gained increasing attention from researchers and scientists over the last two decades. In this article, the nature of the fixed and periodic states are examined for a discrete two-parameter map; a composition of Euler’s numerical map and the logistic map. Further, the dynamical properties such as fixed states, period-doubling, and stability in fixed and periodic states are also described and the onset of chaos is characterized in detail followed by a few lemmas and remarks. Afterward, some scaling methods such as the bifurcation scale, fork-width scale, and Lyapunov exponent are illustrated to examine the appearance of chaos for the discrete two-parameter map. Experimental and numerical simulations are conducted followed by some bifurcation graphs, tables, and remarks. The scaling property is discussed in two key parameters. In addition, a comparative analysis of fork-width length, bifurcation length, and the maximum Lyapunov exponent is also presented to demonstrate the validity of the results.

The Conley–Zehnder indices of the spatial Hill three-body problem

We explore the interaction between the Conley–Zehnder index and bifurcation points of symmetric planar as well as spatial periodic orbits in the spatial Hill three-body problem. We start with the fundamental families of planar periodic orbits which are those of direct (family g) and retrograde periodic orbits (family f). Since the spatial system is invariant under a symplectic involution, whose fixed point set corresponds to the planar problem, planar orbits have planar and spatial Floquet multipliers, and planar and spatial Conley–Zehnder indices. When the Floquet multipliers move through a root of unity, new families of periodic orbits bifurcate and the index jumps. For very low energies, the families g and f arise dynamically from the rotating Kepler problem, and in a recent work (Aydin From Babylonian lunar observations to Floquet multipliers and Conley-Zehnder Indices) we determined analytically their indices. By their numerical continuations for higher energies, we determine the index of various families of planar and spatial periodic orbits bifurcating from g and f. Since these families can bifurcate again and meet each other, this procedure can get complicated. This index leads to a grading on local Floer homology. Since the local Floer homology and its Euler characteristic stay invariant under bifurcation, the index provides important information about the interconnectedness of such families, which we illustrate in form of bifurcation graphs. Since the solutions of Hill’s system may serve as orbits for space mission design or astronomical observations, our results promote the interaction between Symplectic Geometry and practical problems.

Image encryption algorithm based on 1D-SFACF with cross-cyclic shift and adaptive diffusion

In this paper, a new one-dimensional chaotic system is proposed, which is called one-dimensional sine-fractional-adjusted-cosine-fractional (1D-SFACF). The dynamic behavior of the 1D-SFACF is analyzed by LE, bifurcation graph, trajectory, and cobweb plot. Compared to most one-dimensional chaotic systems, the 1D-SFACF has good cryptographic properties, such as larger parameter space and better chaotic behavior. Based on the excellent expressiveness of the 1D-SFACF, we design a secure image encryption algorithm using 1D-SFACF, called the image encryption algorithm based on 1D-SFACF (SFACF-IE). SFACF-IE is divided into four steps. First, the control parameters and initial values of 1D-SFACF are generated through a natural noise using a hash function. Secondly, an adaptive diffusion strategy is proposed. The starting position of the plaintext diffusion is adaptively changed according to the characteristics of the plaintext. Then, use a cross-cyclic shift to scramble. Finally, a hash function is used to generate a feedback key, the scrambled image is the input of the hash function, and the ciphertext is obtained by diffusing again. Experimental evaluations show that the algorithm is resistant to common attacks.

Five-dimensional memristive Hopfield neural network dynamics analysis and its application in secure communication

PurposeThis study aims to improve the complexity of chaotic systems and the security accuracy of information encrypted transmission. Applying five-dimensional memristive Hopfield neural network (5D-HNN) to secure communication will greatly improve the confidentiality of signal transmission and greatly enhance the anticracking ability of the system.Design/methodology/approachChaos masking: Chaos masking is the process of superimposing a message signal directly into a chaotic signal and masking the signal using the randomness of the chaotic output. Synchronous coupling: The coupled synchronization method first replicates the drive system to get the response system, and then adds the appropriate coupling term between the drive The synchronization error and the coupling term of the system will eventually converge to zero with time. The synchronization error and coupling term of the system will eventually converge to zero over time.FindingsA 5D memristive neural network is obtained based on the original four-dimensional memristive neural network through the feedback control method. The system has five equations and contains infinite balance points. Compared with other systems, the 5D-HNN has rich dynamic behaviors, and the most unique feature is that it has multistable characteristics. First, its dissipation property, equilibrium point stability, bifurcation graph and Lyapunov exponent spectrum are analyzed to verify its chaotic state, and the system characteristics are more complex. Different dynamic characteristics can be obtained by adjusting the parameter k.Originality/valueA new 5D memristive HNN is proposed and used in the secure communication

STABILITY ANALYSIS AND NUMERICAL SIMULATIONS OF IVGTT GLUCOSE-INSULIN INTERACTION MODELS WITH TWO TIME DELAYS

This paper presents a systematic study of a mathematical model of glucose and insulin interaction with two time delays, with a focus on analytical studies, bifurcation analysis, and very well numerical simulations. This model based on the Intra-Venous Glucose Tolerance Test (IVGTT) and is presented with two time delays. One delay is the insulin response time to an increase in glucose concentration, and the hepatic glucose production time delay is the other. Then, we establish results on positivity, boundedness, and persistence. We also provide sufficient stability analysis conditions for both local and global asymptotic stability of the proposed models. For the latter, two different strategies are used: stability bifurcation analysis and Lyapunov-Krasovskii functionals. We investigate different regions of parameter space using two approaches, that yield different sets of sufficient conditions for global stability. The bifurcation graphs generated from our extensive and carefully designed simulations complement and confirm these analytical results. The insulin concentration level peaks after the glucose concentration level, according to the numerical simulations.

The dynamics of a memristor-based Rulkov neuron with fractional-order difference

The exploration of the memristor model in the discrete domain is a fascinating hotspot. The electromagnetic induction on neurons has also begun to be simulated by some discrete memristors. However, most of the current investigations are based on the integer-order discrete memristor, and there are relatively few studies on the form of fractional order. In this paper, a new fractional-order discrete memristor model with prominent nonlinearity is constructed based on the Caputo fractional-order difference operator. Furthermore, the dynamical behaviors of the Rulkov neuron under electromagnetic radiation are simulated by introducing the proposed discrete memristor. The integer-order and fractional-order peculiarities of the system are analyzed through the bifurcation graph, the Lyapunov exponential spectrum, and the iterative graph. The results demonstrate that the fractional-order system has more abundant dynamics than the integer one, such as hyper-chaos, multi-stable and transient chaos. In addition, the complexity of the system in the fractional form is evaluated by the means of the spectral entropy complexity algorithm and consequences show that it is affected by the order of the fractional system. The feature of fractional difference lays the foundation for further research and application of the discrete memristor and the neuron map in the future.

Finite-Time Bound Synchronization of the New Chaotic System With Energy Consumption Estimation

As chaotic systems are widely used in many fields, the study of them is becoming more and more in-depth. This paper first presents a new single-equilibrium chaotic system which is only three terms, and some fundamental dynamical feature of the new chaotic system are discussed, such as equilibria, dissipativity, Poincaré diagram, bifurcation graph, etc. Secondly, a new finite-time controllers is designed by using Lyapunov stability theory, and it can be used for bound synchronization of the general chaotic systems. In contrast to the current finite time controller of the chaotic system, the designed controller in this paper does not contain exponential term, it can be simple and eliminate the chattering phenomenon during synchronizationis, which may be easier to implement in practical application. In addition, under the finite time controller, the bound of control energy consumption of the chaotic system is estimated. Finally, the finite-time controllers for the new chaotic system are advanced using the design method of finite-time controller of the general chaotic system, and the result of numerical simulation is given to check its validity by the designed method.

Extreme events in a forced BVP oscillator: Experimental and numerical studies

In this work, we report the existence of extreme events in the well known Bonhoeffer-van der Pol (BVP) oscillator under the excitation of a periodically forced voltage. Extreme events refer to the sudden and random increase in the amplitude of one or more of the state variables of the dynamical system and arise because of the incidence of interior crises or the presence of discontinuous boundaries or intermittency. We have chosen this system because of the fact that it has spawned several systems modelling neuronal dynamics such as Hindmarsh-Rose (HR) and Hodgkin-Huxley (HH) models. Our investigations involve both laboratory experiments and numerical simulations. We have obtained time plots, phase portraits, Poincaré maps, bifurcation graphs, Lyapunov exponents and signal to noise ratio (SNR) to study the general dynamics and to confirm the presence of extreme events, we have used statistical measures such as phase slip analysis, distribution functions for both experimental and numerical data. To the best of our knowledge, we believe that it is for the first time that the occurrence of extreme event has been reported using both real time experimental and numerical studies on this forced BVP system.

Periodic Oscillatory Phenomenon in Fractional-Order Neural Networks Involving Different Types of Delays

This research is chiefly concerned with the stability and Hopf bifurcation for newly established fractional-order neural networks involving different types of delays. By means of an appropriate variable substitution, equivalent fractional-order neural network systems involving one delay are built. By discussing the distribution of roots of the characteristic equation of the established fractional-order neural network systems and selecting the delay as bifurcation parameter, a novel delay-independent bifurcation condition is derived. The investigation verifies that the delay is a significant parameter which has an important influence on stability nature and Hopf bifurcation behavior of neural network systems. The computer simulation plots and bifurcation graphs effectively illustrate the reasonableness of the theoretical fruits.

Nonlinear vibration of a moving flexible printed electron web under multiphysics dynamics

Flexible electronic printing technology is a scientific technology that uses an “ink” material with conductive, dielectric, or semiconductor properties printed on a flexible web substrate to achieve precise preparation of flexible electronic devices, which are widely used in information, energy, medical, and military fields. In the preparation of the printing process of flexible printed electron webs under complex working conditions, the moving web will experience substantial unstable nonlinear dynamic behavior, such as divergence, flutter, bifurcation, and chaos. Accordingly, because of the coupling effects of the complex working conditions of the magnetic field, air and nonlinear electrostatic field forces, it is indispensable to explore the nonlinear dynamic equation of the flexible printed electron web in motion. The theory of multiphysics dynamics establishes a nonlinear vibration equation for the flexible printed electron web under multiphysics conditions. The discrete nonlinear vibration equation of state space equation was obtained by the Bubnov–Galerkin method. Utilizing the Runge–Kutta technique of the fourth-order, Poincaré maps, phase-plane diagrams, power spectra, bifurcation graphs, and time history diagrams of the moving flexible printed electron web were obtained. The influences of the velocity, electrostatic field, magnetic induction intensity, and follower force on the flexible printed electron web were analyzed. In addition, the Ansoft Maxwell finite element simulation software was used to simulate the magnetic field distribution of the moving web during roll-to-roll transmission. This paper determines the stable working range of the moving flexible printed electron web, which provides a theoretical basis for the preparation of flexible printed electronic webs.

Complex Dynamics in a Closed-Loop Supply Chain with Risk Aversion and Fairness Concerns Under Supply Disruption

In this paper, we consider a closed-loop supply chain (CLSC) consisting of two suppliers, one manufacturer, one risk-averse retailer and one fair-caring third-party in the presence of supply disruption. We focus on establishing a dynamic Stackelberg game model with bounded rational expectation and analyzing the game evolution process. The effects of key parameters on the Nash equilibrium solutions and their stability are investigated, as well as the complex dynamical behaviors of the CLSC system are explored by using the stability region, bifurcation graph, the largest Lyapunov exponent (LLE), strange attractors, etc. Moreover, the performance of channel members under different values of parameters is researched by utilizing the (average) expected profits or utilities index. The analysis results reveal that the excessive fast adjustment speed of the manufacturer will lead to the system losing stability and falling into chaos. Also, the retailer’s risk aversion and the third party’s fairness concerns have a destabilization effect on the Nash equilibrium point, while the possibility of supply disruption has different effects on the scope of the adjustment speed of decision variables of the manufacturer. Furthermore, in most cases, an over the top adjustment speed of the manufacturer is disadvantageous to all the channel members for more expected profits, but the third-party can achieve a better performance when the system is in periodic state. Finally, the time-delay feedback control method is proposed to eliminate the system chaos.

Bifurcations in a fractional-order BAM neural network with four different delays

This paper illuminates the issue of bifurcations for a fractional-order bidirectional associative memory neural network(FOBAMNN) with four different delays. On account of the affirmatory presumption, the developed FOBAMNN is firstly transformed into the one with two nonidentical delays. Then the critical values of Hopf bifurcations with respect to disparate delays are calculated quantitatively by establishing one delay and selecting remaining delay as a bifurcation parameter in the transformed model. It detects that the stability of the developed FOBAMNN with multiple delays can be fairly preserved if selecting lesser control delays, and Hopf bifurcation emerges once the control delays outnumber their critical values. The derived bifurcation results are numerically testified via the bifurcation graphs. The feasibility of theoretical analysis is ultimately corroborated in the light of simulation experiments. The analytic results available in this paper are beneficial to give impetus to resolve the issues of bifurcations of high-order FONNs with multiple delays.

Dynamical study on three-species population eco-epidemiological model with fractional order derivatives

The essential target of this work is to analyse the dynamical behaviour of the arbitrary order eco - epidemiological model by adopting the new numerical method. We applied the power law kernel, exponential decay kernel, and generalized Mittag–Leffler kernel functions for treatment of arbitrary order eco-epidemiological model where the considered eco-epidemiological model is a non-linear dynamical system with three population species. Further, we calculated the uniqueness and existence of the solutions by adopting the fixed point hypothesize. Further, we studied the stability analysis for aforesaid model. We examine the possibility for finding new dynamical phase portraits with singular and non-singular arbitrary order operator and demonstrate the dynamical phase portraits at various values of arbitrary order. Furthermore, we have found out the maximal bifurcation graph of the eco-epidemiological system and this eco-epidemiological model is numerically solved by adopting Atangana-Seda (AS) numerical method. We have used Newton’s polynomial.

Chaotic Behaviors in a Nonlinear Game of Two-Level Green Supply Chain with Government Subsidies

In this paper, a two-level green supply chain composed of a manufacturer and a retailer is taken as the background. Considering the consumer’s double consumption preference and the manufacturer’s green product R&D investment, a differential game model of the green supply chain under the government cost subsidy strategy is constructed. Firstly, the equilibrium points of the system are solved and their stability is discussed and analyzed. Secondly, the dynamic evolution process of Nash equilibrium under the parameters of green degree, green preference coefficient, retail channel preference coefficient, coefficient of the sensitivity of price, and adjustment speed are described by numerical simulation. The results show that the two ways of a system entering chaos are Flip bifurcation and N-S bifurcation, respectively, by 2D bifurcation graph, and it is also verified in 1D bifurcation diagram. When the bifurcation parameters are small, the system maintains Nash equilibrium stability. If the green degree of products is increased, the green preference coefficient will also increase; on the contrary, the retail preference coefficient will decrease. Research and development cost subsidy policy can effectively improve the green degree of products and increase the sales volume of products, so as to improve the profit of supply chain members.

Dynamics and Complexity Analysis of Fractional-Order Chaotic Systems with Line Equilibrium Based on Adomian Decomposition

In this paper, the Adomian decomposition method (ADM) is applied to solve the fractional-order system with line equilibrium. The dynamics of the system is analyzed by means of the Lyapunov exponent spectrum, bifurcations, chaotic attractor, and largest Lyapunov exponent diagram. At the same time, through the Lyapunov exponent spectrum and bifurcation graph of the system under the change of the initial value, the influence of fractional order q on the system state can be observed. That is, integer-order systems do not have the phenomenon of attractors coexistence, while fractional-order systems have it.

A new parameter adaptive control of four-dimensional hyperchaotic system

By adding a state feedback controller to a three-dimensional chaotic system, a new four-dimensional hyperchaotic autonomous system is constructed. The nonlinear dynamic behavior of the system is deeply investigated, including power spectrum, Poincare cross section, Lyapunov exponential spectrum and bifurcation graph. The analysis shows that the new four-dimensional hyperchaotic system has complex chaotic motions with different parameters, and the related analog chaotic circuits are designed. Finally, a parameter adaptive controller is designed to achieve global stability control of chaotic systems. The effectiveness of the proposed method is verified by numerical simulation.

Research on Dynamic and Complexity of Energy-Saving Investment about Multichannel and Multienergy Supply Chain

Considering the multienergy structure of the electricity market and supply-side competition reform in China, a dual-channel and multiproduct supply chain model is constructed. There are three players in the game model: new energy company and traditional energy company provide energy for the market and the State Grid at the same time. The State Grid is a retailer who buys electricity from two companies and supplies to the market after converting and transmitting the power. Three companies can invest in grid management for saving energy and reducing losses. The energy loss rate is an exponential function of line loss investment. Through the bifurcation graph, Lyapunov exponent, and the basin of parameter, the complex characteristics of the investment market are analyzed. It is interesting to find the Grazing–Hopf bifurcation which usually occurs in nonlinear circuits. The mixed expectation of bounded rationality and the naive expectation is conducive to suppressing the bifurcation and chaos of the market. When external shocks occur, the control model has good robustness.

Modeling of the CubeSat deployment and initial separation angular velocity estimation

CubeSats applications are recently advanced from purely education tools to actual scientific missions. To meet the high requirements of such scientific missions, the nonlinear deployment dynamics and the initial separation angular velocity from the deployer are required to be investigated and estimated. In this paper, the deployment process is described by single-degree-of-freedom (SDOF) and three-degree-of-freedom (TDOF) vibro-impact models for different deployment stages, and the evolution process from doubling bifurcation to chaos is visualized by the Poincaré section and global bifurcation graph. According to the simulation results, reasonable initial separation angular velocities are consequently estimated. Simultaneously, the largest ratio of the excitation frequency and the natural frequency of the satellite in the deployer is found, which guides the design of CubeSat deployer.

Image encryption based on Henon chaotic system with nonlinear term

Henon chaotic system is a kind of classical chaotic system which is widely studied and applied. However, when we analyze its chaotic characteristics, we can still find some shortcomings. In this paper, a Henon chaotic system with nonlinear terms is proposed and applied to image encryption. Our scheme first modifies the linear term y(n) of the original Henon chaotic system to the nonlinear term ey(n), and then obtains the new Henon chaotic system. Then, a series of methods such as bifurcation graph, Lyapunov exponent, linear complexity, histogram, balance, run characteristics and NIST test were used to verify the performance improvement of the new Henon chaotic system. And then, the random sequences generated by the new Henon chaotic system are combined with the image encryption algorithm to complete the image encryption through three steps: global confusion, bit recombination and pixel value diffusion. Finally, the security of the encryption algorithm is verified by secret key space analysis, anti-statistical attack analysis, anti-differential attack analysis, anti-shearing and noise attack analysis. In this paper, the improvement of Henon chaotic system and the design of encryption algorithm all have achieved good results.

Strategies and stability study for a triopoly game considering product recovery based on closed-loop supply chain

This paper is focused on a triopoly closed-loop supply chain (CLSC) game model, including triopoly oligopolies applying for circular economy development. In the CLSC model, manufacturers sell new products and re-collect waste products for maximizing their profits. This paper analyzes the game processing, the Nash equilibrium solutions, and the stability of the game model. Moreover, the influences of parameters on the system’s stability have been investigated, using the maximal Lyapunov exponent, bifurcation graph, and the stable domain. The results show that the system will be unstable when the adjustment speed of the manufacture is too high; unreasonable price strategies could easily cause system bifurcation; the critical point of system bifurcation is closely related to the collecting price.