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The aim of this work is to analytically investigate the nonlinear dynamic behaviors of a proposed reduced Lorenz system based on q-deformations. The effects of varying the new q-deformation parameter on the dynamical behaviors of the system along with the induced bifurcations of fixed points are explored. In particular, the codimension-one bifurcation analysis is carried out at interior fixed point of the q-deformed system. Explicit conditions for the existence of pitchfork and Neimark–Sacker bifurcations are obtained. Numerical simulations are performed to confirm stability and bifurcation analysis in addition to investigate the effects of variations in system parameters. The changes in system dynamics are explored via the bifurcation diagrams, phase portraits and time series diagrams. Moreover, the quantification of system complex behaviors is depicted through the maximal Lyapunov exponent plots. A cascaded version of the model is proposed to boost its complex dynamics. Then, a chaos-based image encryption scheme, relying on a proposed key-distribution algorithm, is introduced as an application. Finally, several aspects of security analysis are examined for the encryption system to prove its efficiency and reliability against possible attacks.

This work aims at introducing a new discrete fractional order model based on Lotka-Volterra prey-predator model with logistic growth of prey species. The proposed model is a generalization of the standard integer order discrete-time Lotka-Volterra model to its fractional-order version while incorporating also logistic growth for prey population. The equilibrium points of the presented model are firstly obtained, and their stability analysis is conducted. Then, the nonlinear dynamics of the proposed model and possible occurrence of chaotic behavior are explored. The effects of fractional order along with other key parameters in the model are investigated using several techniques. Thorough numerical simulations are carried out where Lyapunov exponents, bifurcation diagrams, phase portraits and as well as $$C_{0}$$ complexity measure are obtained to analyze the dynamics of the proposed model and confirm theoretical results.

This paper focusses on developing two species, where only prey species suffers by a contagious disease. We consider the logistic growth rate of the prey population. The interaction between susceptible prey and infected prey with predator is presumed to be ruled by Holling type II and I functional response, respectively. A healthy prey is infected when it comes in direct contact with infected prey, and we also assume that predator-dependent disease spreads within the system. This research reveals that the transmission of this predator-dependent disease can have critical repercussions for the shaping of prey–predator interactions. The solution of the model is examined in relation to survival, uniqueness and boundedness. The positivity, feasibility and the stability conditions of the fixed points of the system are analysed by applying the linearization method and the Jacobian matrix method.

An efficient scheme for studying multiparametric Zero-Hopf bifurcation is provided to extend A. Nayfeh multiple-time scale technique of codimension-one bifurcations. The suggested approach treats the cases of high codimension bifurcations successfully. Compared to the well-known averaging method, center manifold reduction method and projection method, the present approach is more simple and can be applied with less computational cost and high efficiency to a wider class of problems involving the cases where purely imaginary, zeros and negative real eigenvalues coexist simultaneously.

Reconsidering the Bertrand duopoly game based on the concept of long short-term memory, we construct a fractional-order Bertrand duopoly game by extending the integer-order game to its corresponding fractional-order form. We build such a Bertrand duopoly game, in which both players can make their decisions with long-memory effects. Then, we investigate its Nash equilibria, local stability, and numerical solutions. Using the bifurcation diagram, the phase portrait, time series, and the 0-1 test for chaos, we numerically validate these results and illustrate its complex phenomena, such as bifurcation and chaos.

This paper formulates and analyzes a modified Previte-Hoffman food web with mixed functional responses. We investigate the existence, uniqueness, positivity and boundedness of the proposed model's solutions. The asymptotic local and global stability of the steady states are discussed. Analytical study of the proposed model reveals that it can undergo supercritical Hopf bifurcation. Furthermore, analysis of Turing instability in spatiotemporal version of the model is carried out where regions of pattern creation in parameters space are obtained. Using detailed numerical simulations for the diffusive and non-diffusive cases, the theoretical findings are verified for distinct sets of parameters.

Securing transmission of information between legitimate transmitter and receiver sides is a great challenge for mathematicians, computer scientists and engineers in recent years. This paper aims at achieving three goals. The first of them is to introduce a novel fractional order two dimensional (2D) map having very complex chaotic behavior and distinct large positive values of Lyapunov exponents over wide range of parameters, compared with other 2D maps in literature. Secondly, a new reliable secure encryption scheme combining the associated chaotic pseudo-orbits of the proposed map with the advantages of elliptic curves in public key cryptography is suggested, for first time, and applied to colored images. The hybrid scheme is capable to confirm reliable secret keys exchange in addition to highly obscure and hide transmitted information messages. Finally, a thorough mathematical analysis of security performance and evaluation of encryption scheme immunity against all possible attacks are carried out and proved its efficiency and robustness.

Bifurcation theory (center manifold and Ljapunov-Schmidt reduction, normal form theory, universal unfolding, calculation of bifurcation diagrams) has become an important and very useful means in the solution of nonlinear stability problems in many branches of engineering. The present study deals with qualitative behavior of a two-dimensional discrete-time system for interaction between prey and predator. The discrete-time model has more chaotic and rich dynamical behavior as compare to its continuous counterpart. We investigate the qualitative behavior of a discrete-time Lotka-Volterra model with linear functional response for prey. The local asymptotic behavior of equilibria is discussed for discrete-time Lotka-Volterra model. Furthermore, with the help of bifurcation theory and center manifold theorem, explicit parametric conditions for directions and existence of flip and Hopf bifurcations are investigated. Moreover, two chaos control methods, that is, OGY feedback control and hybrid control strategy, are implemented. Numerical simulations are provided to illustrate theoretical discussion and their effectiveness.

This paper explores a nonlinear Cournot oligopoly with n firms displaying general isoelastic demand. The marginal profits-based gradient rule and the expectation rule Local Monopolistic Approximation were employed in two Cournot oligopoly games. Nash equilibrium stability analysis is carried out on each of the two games to throw light on the effects of demand elasticity and other parameters on the dynamics of the game. Our results show that the influence of demand elasticity on stability depends on firms’ expectation rules.

The main concern of this paper is to discuss stability and bifurcation analysis for a class of discrete predator-prey interaction with Holling type II functional response and harvesting effort. Firstly, we establish a discrete singular bioeconomic system, which is based on the discretization of a system of differential algebraic equations. It is shown that the discretized system exhibits much richer dynamical behaviors than its corresponding continuous counterpart. Our investigation reveals that, in the discretized system, two types of bifurcations (i.e., period-doubling and Neimark–Sacker bifurcations) can be studied; however, the dynamics of the continuous model includes only Hopf bifurcation. Moreover, the state delayed feedback control method is implemented for controlling the chaotic behavior of the bioeconomic model. Numerical simulations are presented to illustrate the theoretical analysis. The maximal Lyapunov exponents (MLE) are computed numerically to ensure further dynamical behaviors and complexity of the model.

In this work, an analytical framework to understand nonlinear dynamics of plasma perturbations model is introduced. In particular, we analyze the model presented by Constantinescu et al. [20] which consists of three coupled ODEs and contains three parameters. The basic dynamical properties of the system are first investigated by the ways of bifurcation diagrams, phase portraits and Lyapunov exponents. Then, the normal form technique and perturbation methods are applied so as to the different types of bifurcations that exist in the model are investigated. It is proved that pitcfork, Bogdanov–Takens, Andronov–Hopf bifurcations, degenerate Hopf and homoclinic bifurcation can occur in phase space of the model. Also, the model can exhibit quasiperiodicity and chaotic behavior. Numerical simulations confirm our theoretical analytical results.

This paper investigates the difference between price and quantity competition in a mixed duopoly game. We describe the behavior of a duopolistic Bertrand competition market with environmental taxes. There are two cases. In the first, the public firm is privatized and in the second, it is not privatized. In case I, private duopoly (postprivatization) where players use different production methods and choose their prices with (bounded rationality and naive). In case II, mixed duopoly (preprivatization) in this case there are two levels for the market including standard objective of the private firm is to maximize profits and including another objective function of the public firm namely “private welfare maximization”. We study numerically the dynamical behaviors of the models. The Nash equilibrium loses stability through a period-doubling bifurcation and the market in the end gets to be disordered. The disordered behavior of the market has been controlled by using feedback control method.

In this paper, a nonlinear quadropoly game based on Cournot model with fully heterogeneous players is established. This game extends the model introduced by Tramontana and Elsadany (Nonlinear Dyn 68:187–193, 2012) who considered a heterogeneous triopoly game with an isoelastic demand function. Here, four different types of players and potentially different marginal costs are considered. Moreover, the assumption of an isoelastic demand function increases the nonlinearity of the final four-dimensional map. The stability of the resulting discrete-time dynamical system is analyzed. The existence of Neimark–Sacker bifurcation near the Nash equilibrium point of the game is shown. Also, based on the Kuznetsov’s normal form technique for discrete-time system, the stability of the Neimark–Sacker bifurcation is also discussed which indicates that the bifurcation is supercritical. Moreover, it is shown that the Nash equilibrium point of the game undergoes period-doubling (flip) bifurcation. Furthermore, the double route to chaotic dynamics in this model, via flip bifurcations and via Neimark–Sacker bifurcation of the Nash equilibrium point, is illustrated. Coexistence of multi-chaotic attractors of the model is illustrated. Simulation tools like bifurcation diagrams, stability regions of parameters, Lyapunov exponent spectrum, phase plots and basins of attraction are used to verify the complex dynamics of the game.

In this paper, a generalized Lotka-Volterra (GLV) system, which is modeled by a set of equations that represents the dynamical behaviors of two-predator and one prey species, is analysed. The boundedness, existence and uniqueness of the solutions of the generalized Lotka-Volterra system are studied. Continuous dependence on initial conditions in the model is also studied to determine the parameter values’ range of the model and time T where the system does not exhibit chaotic behaviors. In addition, extinction scenarios of solutions of the model are discussed. The necessary and sufficient conditions for the asymptotic stability of the equilibrium points of the GLV system are derived. It is also shown that the GLV system undergoes a saddle-node bifurcation and generalized Hopf (Bautin) bifurcation around one of its equilibrium points. The coexisting periodic solutions and chaotic attractors in the system are numerically investigated by phase diagrams and Lyapunov exponent spectrum. Linear feedback control technique is utilized to control the GLV model to its equilibrium points. Moreover, the Lyapunov direct method is used to synchronize two identical chaotic GLV models via linear feedback control technique.

This work is related to dynamics of a discrete-time 3-dimensional plant-herbivore model. We investigate existence and uniqueness of positive equilibrium and parametric conditions for local asymptotic stability of positive equilibrium point of this model. Moreover, it is also proved that the system undergoes Neimark-Sacker bifurcation for positive equilibrium with the help of an explicit criterion for Neimark-Sacker bifurcation. The chaos control in the model is discussed through implementation of two feedback control strategies, that is, pole-placement technique and hybrid control methodology. Finally, numerical simulations are provided to illustrate theoretical results. These results of numerical simulations demonstrate chaotic long-term behavior over a broad range of parameters. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the model.

We consider the model of interaction of predator and prey with omnivore using three different waiting time distributions. The first waiting time is induced by the power law distribution which is the generator of Pareto statistics. The second waiting time is induced by exponential decay law with a particular property of Delta Dirac distribution when the fractional order tends to 1, this distribution is link to the Poison distribution. While the last waiting distribution, induced by the Mittag-Leffler distribution, presents a crossover from exponential to power law. For each model, we presented the conditions under which the existence of unique set of exact solutions is reached using the fixed-point Picard’s method. Making use of a recent suggested numerical scheme, we solved each model numerically and some numerical simulations were generated for different values of fractional orders. We noticed a new attractor which can be considered as a combination of the Brownian motion and power law distribution in the model with the Atangana-Baleanu fractional derivative. With the aim to capture more attractors, we modified the model and presented also some numerical simulations. Our new model provides more attractors than the existing one even for fractional differential cases. We presented finally the Maximal Lyapunov exponent and the bifurcation diagrams. The comparative study shows that modeling with non-local and non-singular kernel fractional derivative leads to more attractors as this kernel is able to capture more physical problems.

A dynamic Cournot duopoly model with isoelastic demand function and knowledge spillover effects is established, in which the goals of two enterprises are relative profit maximization. Since these two enterprises only have incomplete information of the market, then it is assumed that the enterprises are boundedly rational, and they need to adjust their production decision in next period according to the gradient adjustment mechanism. Sufficient conditions for the existence and stable properties of unique Nash equilibrium can be judged with the help of the Jury criterion. The size of stability region is related to the values of parameters and the system has two different paths to chaos, one is via flip bifurcation and another is via Neimark-Sacker bifurcation. Finally, the complex dynamic characteristics also can be further studied by using numerical simulation, including the structures of multi-layer Arnold's tongues and coexistence of attractors.

The positive connection between the total individual fitness and population density is called the demographic Allee effect. A demographic Allee effect with a critical population size or density is strong Allee effect. In this paper, discrete counterpart of Bazykin–Berezovskaya predator–prey model is introduced with strong Allee effects. The steady states of the model, the existence and local stability are examined. Moreover, proposed discrete-time Bazykin–Berezovskaya predator–prey is obtained via implementation of piecewise constant method for differential equations. This model is compared with its continuous counterpart by applying higher-order implicit Runge–Kutta method (IRK) with very small step size. The comparison yields that discrete-time model has sensitive dependence on initial conditions. By implementing center manifold theorem and bifurcation theory, we derive the conditions under which the discrete-time model exhibits flip and Niemark–Sacker bifurcations. Moreover, numerical simulations are provided to validate the theoretical results.

In this paper, the dynamics and bifurcations of a three-species predator-prey model with an omnivore are further investigated. The food web considered in this work comprises prey, predator and a third species, which consumes the carcasses of the predator along with predation of the original prey. The conditions for existence, uniqueness and continuous dependence on initial conditions for the solution of the model are derived. Analytical and numerical bifurcation studies reveal that the system undergoes transcritical and Hopf bifurcations around its equilibrium points. Further, the Hopf bifurcation curves in the parameters’ space along with codimension two bifurcations of equilibrium points and bifurcation of limit cycles that arise in the system are investigated. In particular, the occurrence of generalized Hopf, fold Hopf and Neimarck-Sacker bifurcations is unveiled and illustrates the rich dynamics of the model. Finally, bifurcation diagrams, phase portraits and Lyapunov exponents of the model are presented.

This paper demonstrates dynamics, chaos control, and synchronization in Samardzija-Greller population model with fractional order between zero and two. The fractional-order case is shown to exhibit rich variety of nonlinear dynamics. Lyapunov exponents are calculated to confirm the existence of wide range of chaotic dynamics in this system. Chaos control in this model is achieved via a novel linear control technique with the fractional order lying in (1, 2). Moreover, a linear feedback control method is used to control the fractional-order model to its steady states when 0<α<2. In addition, the obtained results illustrate the role of fractional parameter on controlling chaos in this model. Furthermore, nonlinear feedback synchronization scheme is also employed to illustrate that the fractional parameter has a stabilizing role on the synchronization process in this system. The analytical results are confirmed by numerical simulations.

In this work, the dynamical behavior of fractional-order Hastings–Powell food chain model is investigated and a new discretization method of the fractional-order system is introduced. A sufficient condition for existence and uniqueness of the solution of the proposed system is obtained. Local stability of the equilibrium points of the fractional-order system is studied. Furthermore, the necessary and sufficient conditions of stability of the discretized system are also studied. It is shown that the system’s fractional parameter has effect on the stability of the discretized system which shows rich variety of dynamical behaviors such as Hopf bifurcation, an attractor crisis and chaotic attractors. Numerical simulations show the tea-cup chaotic attractor of the fractional-order system and the richer dynamical behavior of the corresponding discretized system.

Utility function of the form U ( q 1 , q 2 ) = q 1 α + q 2 α is introduced.New dynamical Bertrand games are derived.Stability conditions of equilibrium points are discussed.Chaos and bifurcation behavior are studied. The paper investigates a dynamic Bertrand duopoly with differentiated goods in which boundedly rational firms apply a gradient adjustment mechanism to update their price in each period. The demand functions are derived from an underlying CES utility function. We investigate numerically the dynamical properties of the model. We consider two specific parameterizations for the CES function and study the Nash equilibrium and its local stability in the models. The general finding is that the Nash equilibrium becomes unstable as the speed of adjustment increases. The Nash equilibrium loses stability through a period-doubling bifurcation and the system eventually becomes chaotic either through a series of period-doubling bifurcations or after a Neimark-Sacker bifurcation.

This article investigates the dynamics of a Cournot triopoly game whose demand function is characterized by log-concavity. The game is formed using the bounded rationality approach. The existence and local stability of steady states of the game are analyzed. We find that an increase in the game parameters out of the stability region destabilizes the Cournot–Nash steady state. We confirm our obtained results using some numerical simulation. The simulation shows the consistence with the theoretical analysis and displays new and interesting dynamic behaviors, including bifurcation diagrams, phase portraits, maximal Lyapunov exponent, and sensitive dependence on initial conditions. Finally, a feedback control scheme is adopted to overcome the uncontrollable behavior of the game’s system occurred due to chaos.

In a differentiated triopoly model with heterogeneous firms, the local stability of the Nash equilibrium under both quantity and price competition is analyzed. We find that the presence of a firm following a gradient rule based on marginal profits, and a player with adaptive expectations, determines the local stability of the Nash equilibrium, regardless the competition type, while the effects of the degree of product differentiation on the stability depend on the nature of products. Moreover, the Nash equilibrium is more stable under quantity competition than under price competition.

The delay Cournot duopoly game is studied. Dynamical behaviors of the game are studied. Equilibrium points and their stability are studied. The results show that the delayed system has the same Nash equilibrium point and the delay can increase the local stability region.

This paper presents a new Cournot duopoly game. The main advantage of this game is that the outputs are nonnegative for all times. We investigate the complexity of the corresponding dynamical behaviors of the game such as stability and bifurcations. Computer simulations will be used to confirm our theoretical results. It is found that the chaotic behavior of the game has been stabilized on the Nash equilibrium point by using delay feedback control method.

The dynamical system of four heterogeneous firms is derived. Existence and stability conditions of the fixed points are investigated and also complex dynamics is studied. Numerical simulations are used to illustrate the complex behaviors of the proposed dynamic game. The chaotic behavior of the game has been controlled by using feedback control method.

A nonlinear triopoly game with heterogeneous players is presented. We consider three types of players; boundedly rational, adaptive, and naive. A triopoly game is modelled by a three dimensional discrete dynamical system. The stability conditions of the equilibrium points are analyzed. Numerical simulations are used to show bifurcation diagrams, phase portraits, sensitive dependence on initial conditions and fractal dimension. The chaotic behavior of the model has been stabilized on the Nash equilibrium point, by the use of the Pyragas delay feedback control method.

A discrete-time prey–predator model with Holling type II is investigated. For this model, the existence and stability of three fixed points are analyzed. The bifurcation diagrams, phase portraits and Lyapunov exponents are obtained for different parameters of the model. The fractal dimension of a strange attractor of the model was also calculated. Numerical simulations show that the discrete model exhibits rich dynamics compared with the continuous model, which means that the present model is a chaotic, and complex one.

In this study we investigate the dynamics of a nonlinear discrete-time duopoly game, where the players have heterogeneous expectations. Two players with different expectations are considered; one is boundedly rational and the other thinks with adaptive expectations. The stability conditions of the equilibria are discussed. We show how the dynamics of the game depend on the model parameters. We demonstrate that as some parameters of the game are varied, the stability of Nash equilibrium is lost through period doubling bifurcation. The chaotic features are justified numerically via computing Lyapunov exponents, sensitive dependence on initial conditions and the fractal dimension.

A dynamic Cournot game characterized by players with bounded rationality is modeled by two non-linear difference equations. The stability of the equilibria of the discrete dynamical system is analyzed. As some parameters of the model are varied, the stability of Nash equilibrium is lost and the complex chaotic behavior occurs. Synchronization of two dynamic Cournot duopoly games are considered. In the case of identical players, such dynamical system becomes symmetric, and this implies that synchronized dynamics can be obtained by a simpler one-dimensional model whose dynamics summarizes the common behavior of the two identical players.

A nonlinear dynamical system which describe the time evolution of n-competitors in a Cournot game (Bowley's model) with bounded rationality is analyzed. The existence and stability of the equilibria of this system is studied. The stability conditions of the steady states for two and three players are explicitly computed. Complex behavior such as cycles and chaotic behavior are observed by numerical simulation. Delayed Bowley's with bounded rationality in monopoly is studied. We show that firms using bounded rationality with delay has a higher chance of reaching Nash equilibrium.

This paper presents an analytical framework to investigate the dynamical behavior of a new fractional-order hyperchaotic circuit system. A sufficient condition for existence, uniqueness and continuous dependence on initial conditions of the solution of the proposed system is derived. The local stability of all the system’s equilibrium points are discussed using fractional Routh–Hurwitz test. Then the analytical conditions for the existence of a pitchfork bifurcation in this system with fractional-order parameter less than 1/3 are provided. Conditions for the existence of Hopf bifurcation in this system are also investigated. The dynamics of discretized form of our fractional-order hyperchaotic system are explored. Chaos control is also achieved in discretized system using delay feedback control technique. The numerical simulation are presented to confirm our theoretical analysis via phase portraits, bifurcation diagrams and Lyapunov exponents. A text encryption algorithm is presented based on the proposed fractional-order system. The results show that the new system exhibits a rich variety of dynamical behaviors such as limit cycles, chaos and transient phenomena where fractional-order derivative represents a key parameter in determining system qualitative behavior.

We analyze a nonlinear discrete-time Cournot duopoly game, where players have heterogeneous expectations. Two types of players are considered: boundedly rational and naive expectations. In this study we show that the dynamics of the duopoly game with players whose beliefs are heterogeneous, may become complicated. The model gives more complex chaotic and unpredictable trajectories as a consequence of increasing the speed of adjustment of boundedly rational player. The equilibrium points and local stability of the duopoly game are investigated. As some parameters of the model are varied, the stability of the Nash equilibrium point is lost and the complex (periodic or chaotic) behavior occurs. Numerical simulations is presented to show that players with heterogeneous beliefs make the duopoly game behaves chaotically. Also we get the fractal dimension of the chaotic attractor of our map which is equivalent to the dimension of Henon map., 12 pages, 6figures in one file, accepted in Physica A

This paper analyzes the dynamics of a Cournot duopoly model with different strategies. We offer results on existence, stability and local bifurcations of the equilibrium points. The bifurcation diagrams and Lyapunov exponents of the model are presented to show that the model behaves chaotically with the variation in the parameters. The state variables feedback and parameter variation methods are used to control the chaos of the model.