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Abstract In this paper, we construct a duopoly model of technological innovation based on constant conjectural variation under the assumption of bounded rationality. The equilibrium points of constant conjectural variation and their local stability are investigated. By the theoretical proof, we prove that the model displays Li-Yorke chaos and distributional chaos in the ranges of the output adjusted coefficients by employing the snap-back repeller theory. The technological innovation is the main source of the core competitiveness of a firm, and technological content is an important factor of technological innovation. We find that the firms have the optimal technological contents while they could have the maximal profits in the two different situations, non-cooperative technological innovation and cooperative technological innovation. It's proved that the cooperation of technological innovation can increase the marginal technological content, and benefit to technological progress. Then Numerical simulations are used to illustrate the dynamic analysis of the model under the situations with different technological innovation and conjectural variation. When the output adjusted coefficient changes, a series of complex phenomena including bifurcation, chaos and strange attractor can be observed in our model. And when the technological content is larger, the chaos of the system finally appears. [ABSTRACT FROM AUTHOR]

Highlights • A novel no-equilibrium Jerk-like chaotic system is constructed and explored. • The system generates various asymmetric coexisting hidden attractors. • Transient chaos, periodic bursting oscillation and transient periodic bursting oscillation behaviors are obtained. • The unique phenomenon of one to two full Feigenbaum remerging trees is also observed. • The hardware experimental results are performed to prove the numerical simulations. Abstract The topic associated with hidden attractor and multistability has been received considerable attention recently. In this paper, a novel no-equilibrium Jerk-like chaotic system is constructed and explored. Particularly, owing to the absence of the equilibria, such a new system can be categorized as a system with hidden attractors. More interestingly, this system holds three conspicuous characteristics. The first one is that various asymmetric coexisting hidden attractors and complicated transient chaos behaviors are obtained. The second one is the new finding of the periodic bursting oscillation and unusual phenomenon of transient periodic bursting oscillation in the system. The third one is the observation of the amazing and rare phenomenon of one to two full Feigenbaum remerging trees, namely, antimonotonicity. To the best knowledge of us, the last two special features are first discovered and have never been reported, especially in such no-equilibrium chaotic system that exhibits hidden attractors. With the help of phase portraits, time series, bifurcation diagram, Lyapunov exponents, chaotic dynamical diagram, basin of attraction and so forth, the rich hidden dynamical properties of this system are systematically analyzed and investigated. Additionally, a hardware electronic circuit on a breadboard is carried out. A very good similarity between the hardware experimental results and the theoretical analysis testifies the feasibility and practicality of this original system. [ABSTRACT FROM AUTHOR]

Highlights • Equilibrium points and asymptotic stability. • Uniqueness and existence of solution of the model. • Existence of solutions for 5-D hyperchaotic system with four wings. • Numerical simulations. • Conclusion. Abstract A new 5-D hyperchaotic system with four wings is studied in the light of the newly introduced operator by Atangana and Baleanu with non-local and non-singular fading memory. The basic properties and stability analysis are studied. Picard–Lindelof method is used to examine the existence and uniqueness of solutions of the new 5-D hyperchaotic system with four wings. The numerical simulation results depict a new chaotic behaviours with the ABC numerical scheme. [ABSTRACT FROM AUTHOR]

In this contribution, a new four-wing chaotic system with a smooth piecewise quadratic nonlinearity is introduced. The novel system is inspired from the cubic one introduced by Sampath et al. (2015). Fundamental properties of the new system are discussed and its complex behaviors are characterized using classical nonlinear diagnostic tools. This system exhibits a rich repertoire of dynamic behaviors when suitable set of parameters are chosen including a single two-wing and four-wing chaotic attractors. Further analysis of the novel system shows that four disconnected coexisting stable states can be observed with different initial values. Moreover, the smooth quadratic nonlinearity is easily implemented by using off-the-shelf electronic components (instead of analog multiplier ship in the case of a cubic nonlinearity). The synchronization and the feasibility of the proposed mathematical model are also presented by developing an adaptive synchronization scheme of two identical four-wing chaotic systems and using PSpice simulations based on an electronic analog of the model. [ABSTRACT FROM AUTHOR]

In the real life, we often simultaneously encounter various social dilemmas, which are also inclined to be voluntarily participated in, instead of previous assumption's compulsory participation in. Accounting on this realistic scenario, we have introduced the mechanism that the individuals have access to different payoff matrices corresponding to different social dilemmas to participate in the multigame with three strategies to choose, including cooperation, defection, going it alone. Furthermore, we set a proportion ψ /2 of the population to play the Prisoner's Dilemma, a proportion ψ /2 of the population to play Snowdrift and a proportion 1 − ψ of the population to play the weak Prisoner's Dilemma, which results in the fact that the mean payoff matrix returns to the basic weak PD. Though numerical simulations, we find that for the smaller temptation to defect, the cooperation can be enhanced by the diversity of the sucker's payoff in the multigame contrast to the basic case. In addition, when the contribution of sucker's payoff is larger or more players choose to play the Prisoner's Dilemma and Snowdrift, the cooperators become more dominated. [ABSTRACT FROM AUTHOR]

This research investigates pattern self-organization along the route to chaos in a space- and time-discrete predator–prey system, where the prey shows convection movement in space. Through analysis on Turing instability of the system, pattern self-organization conditions are determined. Based on the conditions, simulations are performed under two initial conditions, demonstrating two pattern transitions along the route to chaos. In the first pattern transition, the patterns start from regular stripes, experiencing twisted stripes, then return to regular stripes again. The second pattern transition is much more complex and shows three stages. Especially, an alternation between ordered patterns and disordered chaos is found, contributing greatly to the spatiotemporal complexity of the system. When the system stays at the homogeneous chaotic states, Turing instability driven by convection and diffusion can still force the self-organization of regular striped patterns. The finding in this research provides a new comprehending for pattern self-organization and transition in spatially extended predator–prey systems. [ABSTRACT FROM AUTHOR]

In this paper, a fuzzy method is combined with intermittent control method to realize synchronization of chaotic system. Two plant rules of intermittent control are considered to get two theorems. Fuzzy scheme for synchronization is proposed in theorem. Finally, a simulation example is proposed to verify the effectiveness of our results. [ABSTRACT FROM AUTHOR]

Based on the generalized Hamiltonian system, a new method for constructing a class of three-dimensional (3-D) chaotic systems is presented in this paper. After choosing the proper parameters of skew-symmetric matrix, dissipative matrix and external input, one smooth 3-D chaotic system is proposed to show the effectiveness of the proposed method. Numerical simulation techniques, including phase portraits, Poincaré sections, Lyapunov exponents and bifurcation diagram, illustrate that the proposed 3-D system has periodic, quasi-periodic and chaotic flows under the conditions of different parameters. [ABSTRACT FROM AUTHOR]

In this study, the dynamical behaviors of a discrete time eco-epidemiological system are discussed. The local stability, bifurcation and chaos are obtained. Moreover, the global asymptotical stability of this system is explored by an iteration scheme. The numerical simulations illustrate the theoretical results and exhibit the complex dynamical behaviors such as flip bifurcation, Hopf bifurcation and chaotic dynamical behaviors. Our main results provide an efficient method to analyze the global asymptotical stability for general three dimensional discrete systems. [ABSTRACT FROM AUTHOR]

Using both analytical and numerical methods, global dynamics including subharmonic bifurcations and chaotic motions for a class of inverted pendulum system are investigated in this paper. The expressions of the heteroclinic orbits and periodic orbits are obtained analytically. Chaos arising from heteroclinic intersections is studied with the Melnikov method. The critical curves separating the chaotic and non-chaotic regions are obtained. The conditions for subharmonic bifurcations are also obtained. It is proved that the system can be chaotically excited through finite subharmonic bifurcations. Some new dynamical phenomena are presented. Numerical simulations are given, which verify the analytical results. [ABSTRACT FROM AUTHOR]

This paper addresses the stabilization problem of a class of fractional order chaotic systems. The analytically obtained control structure, derived by blending the systematic backstepping procedure with Mittag-Leffler and Lyapunov stability results, helps in obtaining stability of a special case of strict feedback class of fractional order chaotic systems and at the same time avoids the singularity problem. The stabilizing controller is derived for a class of three dimensional systems which can be expressed in strict-feedback form. Thereafter, the methodology has been applied to two example systems i.e. chaotic Lorenz system and Lü system belonging to the addressed class to show the application of results. Numerical simulation results given at the end confirm the efficacy of the scheme presented here. [ABSTRACT FROM AUTHOR]

In the present paper, the synchronization by the Ge-Yao-Chen (GYC) partial region stability theory of chaotic Mathieu-Van der Pol and chaotic Duffing-Van der Pol systems with fractional order-derivative is proposed. Numerical simulations show that this synchronization technique is very effective and it turns out that the fractional order-derivative induces quick synchronization compared to integer order-derivative of these systems. In order to bring out the chaotic behavior of these systems either with fractional or with integer order-derivative, we simulate their phase portraits and the Lyapunov exponent. Moreover, we provide in this work an approximated solution to both systems to show that the solution of such a system can be represented as a simple power-series function. Furthermore, the representation of the error dynamics with respect to the time before and after the control action approves the effectiveness of the control method and proves the possibility of stabilization and controllability of chaotic systems with an appropriate. Furthermore, the synchronization of the fractional Mathieu-Van der Pol system using the fractional Duffing-Van der Pol system is simulated. [ABSTRACT FROM AUTHOR]

We report the dynamics of a low dimensional fractional order forced LCR circuit using Chua’s diode. The stability analysis is performed for each segment of the piecewise linear curve of Chua’s diode and the conditions for the oscillation and double scroll chaos are obtained. The effect of fractional order on the bifurcation points are studied with the help of bifurcation diagrams. We consider both the derivatives of the systems current and voltage as fractional derivatives. When the order of the derivatives is decreased, the system exhibits interesting dynamical behavior. For instance, the value of the fractional order corresponding to the voltage is decreased, the chaotic regime in the system decreases. But in the case of current, the chaotic regime in the system increases initially and beyond a certain value of order, the chaotic regime decreases and extinguishes from the system. We found the lowest order for exhibiting chaos in the fractional order of the circuit as 2.1. For the first time, the experimental analogue of our proposed system is made by using the frequency domain approximation. The results are obtained from the experimental observations are compared with numerical simulations and found that they match closely with each other. The existence of chaos in the circuit is analyzed with the help of 0-1 test and power spectrum. [ABSTRACT FROM AUTHOR]

By utilizing a memristor to substitute a coupling resistor in the realization circuit of a three-dimensional chaotic system having one saddle and two stable node-foci, a novel memristive hyperchaotic system with coexisting infinitely many hidden attractors is presented. The memristive system does not display any equilibrium, but can exhibit hyperchaotic, chaotic, and periodic dynamics as well as transient hyperchaos. In particular, the phenomenon of extreme multistability with hidden oscillation is revealed and the coexistence of infinitely many hidden attractors is observed. The results illustrate that the long term dynamical behavior closely depends on the memristor initial condition thus leading to the emergence of hidden extreme multistability in the memristive hyperchaotic system. Additionally, hardware experiments and PSIM circuit simulations are performed to verify numerical simulations. [ABSTRACT FROM AUTHOR]

A discrete predator-prey system with square root functional response is presented. We study the existence and local stability analysis of the system. The conditions of existence of flip and Niemark-Sacker bifurcations in the system are derived. Furthermore, the chaotic behavior of the system in the sense of Marotto is proved. Numerical simulations are performed to show the consistence with analytical results and also to exhibit the complexity of the system. Finally, chaos control in the system is achieved via OGY feedback control method. [ABSTRACT FROM AUTHOR]

The analysis of circuit employing the Kirchhoff's circuit laws also known as dynamics of Chua's circuit is extended in this work using the newly established fractional derivative with nonlocal and non-singular kernel. A new numerical analysis is presented and used to solve the extended model. Some numerical simulations are done for different values of the fractional order and new chaotic behaviors are obtained. [ABSTRACT FROM AUTHOR]

Stabilizing an unstable equilibrium point in a fractional-order chaotic system is studied using prediction-based feedback method in this paper. Sufficient conditions for stabilization of equilibrium points are explicitly given and rigorously proven. The proposed approach is effective and easy to be implemented, but its application is limited to systems with no external disturbances and uncertainties in the model description. Numerical simulations show the effectiveness of the method. [ABSTRACT FROM AUTHOR]

Based on the observance in human society, the satisfaction level of an individual as a result of an obtained payoff depends on personal tendency to some extent; we establish a new model for spatial prisoner’s dilemma games. We describe individual satisfaction as a stochastically deviated value around each of the four payoffs stipulated by a payoff matrix, which is maintained throughout the life of a certain agent. When strategy updating, an agent who refers to his own satisfaction level cannot see neighbors’ satisfaction levels but can only observe neighbors’ accumulated payoffs. By varying the update rule and underlying topology, we perform numerical simulations that reveal cooperation is significantly enhanced by this change. We argue that this enhancement of cooperation is analogous to a stochastic resonance effect, like the payoff noise effects Perc (2006). [ABSTRACT FROM AUTHOR]

This paper aims at studying local and global dynamics in a nonlinear duopoly with quantity-setting firms and non-cooperative advertising investments that affect the degree of (horizontally) differentiated products. It concentrates on persuasive advertising in a model where each firm has limited information and uses a behavioural rule to set the quantity for the subsequent period. By using some mathematical techniques and numerical simulations, our results show the existence of weak (à la Milnor) attractors, multistability and chaotic dynamics. In the long term, firms may continuously shift from states in which they invest in advertising to states in which advertising investment is absent. [ABSTRACT FROM AUTHOR]

This paper considers a mixed duopoly competition model, in which two enterprises produce heterogeneous commodities and compete for output in the same market. One enterprise separates the ownership from the management rights, and the enterprise owner entrusts a manager to manage the enterprise. The other enterprise considers consumer surplus. Through stability theory and numerical simulation, the local stability of each equilibrium point and the way the system performs bifurcation are analyzed. By using the basin of attraction, the global stability of the system is studied. The multistability is explained and the properties and some phenomena of the basin of attraction and the chaotic attractor are analyzed by using the knowledge of noninvertible map and critical curve. Finally, the influence of parameters on enterprise profit and objective is also analyzed, which is helpful for enterprises to make decisions in the process of game. • A mixed duopoly Cournot competition model is established based on heterogeneous products. • The complex dynamic behavior of the economic system is analyzed by means of chaos theory and numerical simulation. • The complex structures of the basin of attraction and chaotic attractor are studied by using noninvertible map. • The influence of the parameters on the average objective and the average profit of the two enterprises is analyzed. [ABSTRACT FROM AUTHOR]

In this paper, chaotic dynamics of the vibro-impact system under bounded noise excitation is investigated by an extended Melnikov method. Firstly, the Melnikov method in the deterministic vibro-impact system is extended to the stochastic case. Then, a typical stochastic Duffing vibro-impact system is given to application. The analytic conditions for occurrence of chaos are derived by using the random Melnikov process in the mean-square-value sense. In addition, the numerical simulations confirm the validity of analytic results. Also, the influences of interesting system parameters on the chaotic dynamics are discussed. [ABSTRACT FROM AUTHOR]

In Aziz-Alaoui (2002) a three species ODE model, based on a modified Leslie–Gower scheme is investigated. It is shown that under certain restrictions on the parameter space, the model has bounded solutions for all positive initial conditions, which eventually enter an invariant attracting set. We show that this is not true. To the contrary, solutions to the model can blow up in finite time, even under the restrictions derived in Aziz-Alaoui (2002), if the initial data is large enough. We also prove similar results for the spatially extended system. We validate all of our results via numerical simulations. [ABSTRACT FROM AUTHOR]

• The nonlinear chaotic complex system are studied. • The ultimate bound of the nonlinear chaotic complex systems are presented. • The obtained solutions are presented by 2D figures. The competitive modes for a nonlinear chaotic complex system are studied in this paper. In hyperchaotic, chaotic, and periodic cases, we examined competitive modes that are a tool for detecting chaos in a system. Also, using an analytical method and Lagrange optimization, we were able to calculate the ultimate bound of the nonlinear chaotic complex systems. We have presented is simpler and more accurate than other methods that implicitly calculate the ultimate bound. The estimation of the explicit ultimate bound can be used to study chaos control and chaos synchronization. Numerical simulations illustrate the analytical results. [ABSTRACT FROM AUTHOR]

The study of chaotic PDEs with variable coefficients involves higher level of complexity than for the ones with constant coefficients. In this paper, we investigate the chaotic oscillations of a one-dimensional second order linear hyperbolic PDE with variable coefficients that is factorizable as a product of two noncommutative first order operators and the boundary conditions at both ends of the PDE are general nonlinear. Numerical simulations are provided to illustrate the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]