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This paper examines a class of two-species reaction-diffusion-advection competition models with two time delays. A system of DDE equations was derived, both theoretically and numerically, using the Galerkin technique method. A condition is defined that helps to find the existence of Hopf bifurcation points. Full diagrams of the Hopf bifurcation points and areas of stability are investigated in detail. Furthermore, we discuss three different sources of delay on bifurcation maps, and what impacts of all these cases of delays on others free rates on the regions of the Hopf bifurcation in this model. We find two different stability regions when the delay time is positive (τ > 0), while the no-delay case (τ = 0) has only one stable region. Moreover, the effect of delays and diffusion rates on all free others parameters in this model have been considered, which can significantly impact upon the stability regions in both population concentrations. It is also found that, as diffusion increases, the time delay increases. However, as the delay maturation is increased, the Hopf points for both proliferation of the population and advection rates are decreased and it causes raises to the region of instability. In addition, bifurcation diagrams are drawn to display chosen instances of the periodic oscillation and two dimensional phase portraits for both concentrations have been plotted to corroborate all analytical outputs that investigated in the theoretical part. • To study the effect of diffusion with two different delay terms on the two-species advection competition system. • To drive theoretical outcomes of the DDE obtainable utilizing the Galerkin technique tool. • To find, theoretically, a condition that assists to obtain the existence of Hopf bifurcation points. • To construct the bifurcation diagram for the DDE and DPDE systems via numerical simulation to confirm theoretical results. [ABSTRACT FROM AUTHOR]

• Incorporating time delay has a considerable impact on the dynamics of prey– predator interactions with fading memory and Allee effect. • Different values of time delays in competition and cooperation can lead to or eliminate chaos. • Predator density depends not only on the current density of prey but also on the recent past density of prey. • The stability of the system switches from stable (unstable) to unstable (stable) through Hopf bifurcation in the presence of time delays. This paper deals with a model of population dynamics comprising Allee effect and weakening memory with constant time delays. Since predator density depends on the prey density in current time and in past, a two-component model of prey-predator interactions is complemented with a third differential equation for the influence of recent past. The role of constant time delays incorporated in the functional form of Allee effect (referred as delays in competition and cooperation) is investigated analytically and numerically. Steady states of the model are obtained and the local stability analysis around the coexisting state is calculated in the presence of both delays. The critical threshold for time delays, above which the stability of the system switches from stable (unstable) to unstable (stable), is computed for various cases. Analytical findings of this paper are supported with numerical simulations, where time evolution as well as numerical bifurcation diagrams are presented. The results of this paper demonstrate that the influence of past on the prey-predator density in the present of time delay may have a considerable effect upon the system behaviour and can give important insights into underlying biological mechanism. [ABSTRACT FROM AUTHOR]

Highlights • A new compartmental model for the co-dynamics of measles and dysentery diarrhea is proposed. • The system exhibits a backward bifurcation. • Optimal control and cost-effectiveness analysis are addressed. • Numerical simulations illustrate theoretical results and indicate suggestions for disease elimination. • The most cost-effective strategy for the control of the diseases is recommended. Abstract In this paper we propose a co-dynamics deterministic system for measles and dysentery diarrhea diseases in a single host population. Using center manifold theory, we show that the co-dynamics model may exhibit a backward bifurcation for some parameter values. Numerical simulations of the system show that the two diseases always coexist if R 0 > 1. The system is extended to include time-dependent control-variables: vaccination, treatment and sanitation of the environment, to minimize the number of infected humans and the cost of implementation of the controls. The Pontryagin Maximum Principle was employed to find the necessary conditions for the existence of the optimal controls. The numerical simulations show that the effective controls could reduce the diseases in the community. The incremental cost-effectiveness ratio was used to quantify the cost-effectiveness analysis. It is found that the control strategy which implements vaccination, treatment of dysentery diarrhea and sanitation of the environment is the most cost-effective. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a dynamical two-stage Cournot duopoly game with R&D spillover effect and product differentiation is established. The stability of all the equilibrium points is studied using Jury criterion, and then the stability condition is given. The direction of flip bifurcation is given by using central manifold theorem and norm form theory. By numerical simulation, two routes to chaos are studied through 2-D bifurcation diagram, one of which is flip bifurcation and the other is Neimark-Sacker bifurcation at period-2 point. The system will lose its stability as the speed of adjustment increases. In addition, a common nonlinear phenomenon named intermittent chaos is observed in the built model. Two types of intermittent chaos are displayed in time series plots and phase diagrams, one of which is PM-III intermittent chaos and the other is induced by crisis. And then four types of coexistence of attractors are illustrated through basin of attraction, which are coexistence of periodic attractor and chaotic attractor, coexistence of multiple chaotic attractors, coexistence of periodic attractor, quasi-periodic attractor and chaotic attractor, and coexistence of multiple periodic attractors, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we consider theoretical and numerical properties of a nonlinear boundary-value problem which is strongly related to the well-known Gelfand–Bratu model with parameter λ . When approximating the nonlinear term in the model via a Taylor expansion, we are able to find new types of solutions and multiplicities, depending on the final index N in the expansion. The number of solutions may vary from 0, 1, 2 to ∞. In the latter case of infinitely many solutions, we find both periodic and semi-periodic solutions. Numerical experiments using a non-standard finite-difference (NSFD) approximation illustrate all these aspects. We also show the difference in accuracy for different denominator functions in NSFD when applied to this model. A full classification is given of all possible cases depending on the parameters N and λ . [ABSTRACT FROM AUTHOR]

The memristor has drawn a considerable interest when the nanoscale memristor is regarded as the critical element of novel ultra-high density and low-power non-volatile memories. The nonlinearity of electric circuit is enhanced and the dynamical behavior becomes more complex when memristor is used in circuits because it memductance is dependent on the inputs current. Josephson Junction (JJ) coupled resonator also present complex dynamical behaviors in nonlinear circuit because JJ is used as sensitive inductive component. The Josephson Junction circuit employing memristor is designed in this paper. Firstly, dynamical properties about this model are discussed by numerically calculating phase portraits, Lyapunov exponents and bifurcation diagram. It is found that appropriate parameters setting can induce distinct chaotic and periodical states by analyzing the output series. The dynamical response and potential mechanism for behavior selection is discussed. Interestingly, the chaos encryption based on Josephson junction circuit coupled by memristor is investigated as well. [ABSTRACT FROM AUTHOR]

• To demonstrate the Galerkin method's validity, via the accuracy of its predictions using DPDE models. • To structure a process for applying the theoretical framework to find theoretical results. • To show the impacts of diffusion and immune response delay on the model, through bifurcation diagrams. • To determine the steady-state solutions, Hopf points, and the stable and unstable zones. • To illustrate comparisons of the DDE and DPDE approaches through a selection of 2-D and 3-D solution maps. This paper studies a diffusive viral infection system with delayed immune response in the one-domain system. A system of DDE equations was explored, both analytically and numerically, using the Galerkin method. A condition that helps to find Hopf bifurcation points is determined. Full maps of the Hopf bifurcation points as well regions of stability are constructed and considered in detail. It is shown that the time delay of cytotoxic T lymphocyte (CTL) response and the diffusion parameter can significantly impact upon the stability regions. Furthermore, the influences of the other free values have been examined for their effects on stability. It is found that, as diffusion increases, the CTL response delay increases, and also as the CTL response delay is increased, the Hopf points for both generation rate and activate rate are decreased, whereas the Hopf points for the infection and death rates increased. Moreover, an increase diffusion results in an increase in the Hopf points for growth rate and activation rate, while the Hopf bifurcations are decreased for the death rate of infected cells. Bifurcation diagrams are plotted to show selected examples of limit cycle behavior (periodic oscillation), and 3-D solutions for the three concentrations in the model have been plotted to corroborate all analytical results from the theoretical section. [ABSTRACT FROM AUTHOR]

• A novel fractional q-deformation chaotic map is constructed. • Chaotic attractor and strange nonchaotic attractor is reported in the novel fractional q-deformation chaotic map. • the novel fractional q-deformation chaotic map has better chaotic properties. • the fractional version of q-deformation chaotic map adds two new degrees of freedom, which will greatly improve the application value of this system. In this paper, the dynamical behaviors of fractional q-deformation chaotic map are analyzed. Firstly, the fractional q-deformation chaotic map is proposed by employing the Caputo delta difference operator. Secondly, the rich dynamical behaviors, such as numerically stable period (NSP) attractor, quasi-periodic attractor, strange nonchaotic attractor, and chaotic attractor, of the proposed map are discussed by utilizing bifurcation diagram, phase diagram, and 0–1 test. Thirdly, two controllers are designed to study the chaos control and synchronization of the fractional q-deformation chaotic map. Finally, numerical simulations are presented to demonstrate the findings. [ABSTRACT FROM AUTHOR]

• FFT bifurcation as a tool for investigating complex dynamics. • Studying discrete-time and continuous-time systems. • Comparing FFT bifurcation diagrams with the conventional bifurcation diagrams. • The FFT bifurcation shows interesting results in comparison to conventional bifurcation diagrams. This paper presents FFT bifurcation as a tool for investigating complex dynamics. Firstly, two well-known chaotic systems (Rössler and Lorenz) are discussed from the frequency viewpoint. Then, both discrete-time and continuous-time systems are studied. Various systems with different properties are discussed. In discrete-time systems, Logistic map and a biological map are investigated. For continuous-time systems, a system with a stable equilibrium, forced van der Pol system, and a system with a line of equilibria are discussed. For each system under investigation, the proposed FFT bifurcation diagrams are compared with the conventional bifurcation diagrams, showing some interesting information uncovered by the FFT bifurcation. For periodic trajectories, the FFT bifurcations show high power at the dominant frequency and harmonics. By doubling the periods, their dominant frequencies are halved, and more harmonics emerge in the studied frequency intervals. For the chaotic dynamics, the FFT bifurcation shows a wideband power spectrum. The FFT bifurcation shows interesting results in comparison to conventional bifurcation diagrams. [ABSTRACT FROM AUTHOR]

The evolution of bursting oscillations in a parametrically excited dynamical system with order gap between the excited frequency and the natural frequency is investigated in this paper. By regarding the periodic excited term as a slow-varying parameter, different forms of bifurcations of the system are obtained. Base on the overlap between the bifurcation diagram and the phase portrait, the mechanism of different types of bursting oscillations are obtained. Furthermore, some phenomena in bursting oscillations such as symmetry breaking behavior are explained through the bifurcations occurring at the transitions between the quiescent state (QS) and spiking state (SP). [ABSTRACT FROM AUTHOR]

This paper considers a basic model for a spread of two diseases in a population. The equilibria of the model are found, and their stability is investigated. In particular, we prove the stability result for a disease-free and a one-disease steady-states. Bifurcation diagrams are used to analyse the stability of possible branches of equilibria, and also they indicate the existence of a co-infected equilibrium with both diseases present. Finally, numerical simulations of the model are performed to study the behaviour of the solutions in different regions of the parameter space. [Copyright &y& Elsevier]

• In this paper, we present a new fractional-order hyperchaotic memristor oscillator. • The proposed system has been investigated through numerical simulations. • A novel robust adaptive controller is implemented to synchronize the system. • As an application, the system has been implemented for voice encryption. The present study proposes a new fractional-order hyperchaotic memristor oscillator. The proposed system is studied through numerical simulations and analyses, such as the Lyapunov exponents, bifurcation diagrams, and phase portraits. Then, using the sliding mode concept, a robust adaptive control scheme is designed to synchronize the proposed system. The adaptation mechanism is implemented to estimate the unknown parameters of the slave system. Then, the output of the proposed adaptation mechanism is used for the control scheme. The stability of the closed-loop system is proven via a fractional version of the Lyapunov stability theorem and Barbalat's lemma. Numerical simulations of synchronization are shown to investigate the performance of the developed control technique on the uncertain fractional-order hyperchaotic memristor oscillator. Finally, as an engineering application, the proposed fractional-order system is implemented for voice encryption. In this regard, to show the appropriate performance of the proposed system for voice encryption, statistical characteristic of the encryption and decryption processes are performed through different methods including correlation, entropy, root mean square, and root sum of squares. [ABSTRACT FROM AUTHOR]

• Turing and Hopf bifurcations are studied for the predator-prey model. • Weakly nonlinear analysis is employed to identify pattern selection. • Various patterns in two-dimensional space are presented and explained. The Turing and Turing-Hopf bifurcations of a Leslie-Gower type predator-prey system with ratio-dependent Holling III functional response are investigated in this paper. Complex and interesting patterns induced by the bifurcations are identified theoretically and numerically. First the existence conditions of the Turing instability and the Turing-Hopf bifurcation are established from the theoretical analysis, respectively. Then by employing the technique of weakly nonlinear analysis, amplitude equations generated near the Turing instability critical value are derived. Various spatiotemporal patterns, such as homogeneous stationary state patterns, hexagonal patterns, coexisting patterns, stripe patterns, and their stability are determined via analyzing the obtained amplitude equations. Numerical simulations are presented to illustrate the theoretical analysis. Especially, the analogous-spiral and symmetrical wave patterns can be found near the codimension-two Turing-Hopf bifurcation point. A security center of the prey species can be found as well. These spatiotemporal patterns are explained from the perspective of the predators and prey species. [ABSTRACT FROM AUTHOR]

• Pitchfork bifurcation in the case of rigid boundaries is reported. • A new Fourier-Galerkin expansion is proposed for rigid boundaries. • Chaotic, mildly chaotic and periodic motions are reported. • Results of rigid boundaries are compared with those of free boundaries. A non-linear stability analysis of Rayleigh-Bénard convection is reported in the paper for the case of rigid isothermal boundaries. The Ginzburg-Landau equation with cubic non-linearity is obtained. Out of the three fixed points of the equation, one arises in the pre-onset regime and two during post-onset. The pre-onset one is unstable at post-onset times and hence we have a pitchfork bifurcation point. Comparing heat transports it is found that the Nusselt number of the rigid isothermal boundary condition is less than that of the free isothermal case. Prandtl number influence on the Nusselt number is negligible in very high Prandtl number liquids at all times and at long times this is true for all liquids. Using vital information from the procedure leading to the Ginzburg-Landau equation a minimal Fourier-Galerkin expansion is proposed to derive a generalized Lorenz model for rigid-isothermal boundaries. The Hopf bifurcation point signaling onset of chaos is obtained by transforming the Lorenz model of rigid isothermal boundaries to a standard form resembling the classical one. The possibility of regular, chaotic, periodic and mildly chaotic motions in the Lorenz system is shown through the plots of the maximum Lyapunov exponent and the bifurcation diagram. [ABSTRACT FROM AUTHOR]