Mechanism analysis of regulating Turing instability and Hopf bifurcation of malware propagation in mobile wireless sensor networks

A dynamical model is constructed to depict the spatial-temporal evolution of malware in mobile wireless sensor networks(MWSNs). Based on such a model, we design a hybrid control scheme combining parameter perturbation and state feedback to effectively manipulate the spatiotemporal dynamics of malware propagation. The hybrid control can not only suppress the Turing instability caused by diffusion factor but can also adjust the occurrence of Hopf bifurcation induced by time delay. Numerical simulation results show that the hybrid control strategy can efficiently manipulate the transmission dynamics to achieve our expected desired properties, thus reducing the harm of malware propagation to MWSNs.

Hopf bifurcation and phase synchronization in memristor-coupled Hindmarsh–Rose and FitzHugh–Nagumo neurons with two time delays

A memristor-coupled heterogenous neural network consisting of two-dimensional(2D)FitzHugh–Nagumo(FHN)and Hindmarsh–Rose(HR)neurons with two time delays is established.Taking the time delays as the control parameters,the existence of Hopf bifurcation near the stable equilibrium point in four cases is derived theoretically,and the validity of the Hopf bifurcation condition is verified by numerical analysis.The results show that the two time delays can make the stable equilibrium point unstable,thus leading to periodic oscillations induced by Hopf bifurcation.Furthermore,the time delays in FHN and HR neurons have different effects on the firing activity of neural network.Complex firing patterns,such as quiescent state,chaotic spiking,and periodic spiking can be induced by the time delay in FHN neuron,while the neural network only exhibits quiescent state and periodic spiking with the change of the time delay in HR neuron.Especially,phase synchronization between the heterogeneous neurons is explored,and the results show that the time delay in HR neurons has a greater effect on blocking the synchronization than the time delay in FHN neuron.Finally,the theoretical analysis is verified by circuit simulations.

Hopf bifurcation of nonlinear system with multisource stochastic factors

The article mainly explores the Hopf bifurcation of a kind of nonlinear system with Gaussian white noise excitation and bounded random parameter.Firstly,the nonlinear system with multisource stochastic fac-tors is reduced to an equivalent deterministic nonlinear system by the sequential orthogonal decomposi-tion method and the Karhunen-Loeve(K-L)decomposition theory.Secondly,the critical conditions about the Hopf bifurcation of the equivalent deterministic system are obtained.At the same time the influence of multisource stochastic factors on the Hopf bifurcation for the proposed system is explored.Finally,the theorical results are verified by the numerical simulations.

Existence of Supercritical Hopf Bifurcation on a Type-Lorenz System

In this paper, a system of Lorenz-type ordinary differential equations is considered and, under some assumptions about the parameter space, the presence of the supercritical non-degenerate Hopf bifurcation is demonstrated. The technical tool used consists of the Central Manifold theorem, a well-known formula to calculate the Lyapunov coefficient and Hopf’s Theorem. For particular values of the parameters in the parameter space established in the main result of this work, a graph is presented that describes the evolution of the trajectories, obtained by means of numerical simulation.

Hopf Bifurcation of Nonresident Computer Virus Model with Age Structure and Two Delays Effects

This paper constructed and studied a nonresident computer virus model with age structure and two delays effects. The non-negativity and boundedness of the solution of the model have been discussed, and then gave the basic regeneration number, and obtained the conditions for the existence and the stability of the virus-free equilibrium and the computer virus equilibrium. Theoretical analysis shows the conditions under which the model undergoes Hopf bifurcation in three different cases. The numerical examples are provided to demonstrate the theoretical results.

Bifurcation analysis and control study of improved full-speed differential model in connected vehicle environment

In recent years, the traffic congestion problem has become more and more serious, and the research on traffic system control has become a new hot spot. Studying the bifurcation characteristics of traffic flow systems and designing control schemes for unstable pivots can alleviate the traffic congestion problem from a new perspective. In this work, the full-speed differential model considering the vehicle network environment is improved in order to adjust the traffic flow from the perspective of bifurcation control, the existence conditions of Hopf bifurcation and saddle-node bifurcation in the model are proved theoretically, and the stability mutation point for the stability of the transportation system is found. For the unstable bifurcation point, a nonlinear system feedback controller is designed by using Chebyshev polynomial approximation and stochastic feedback control method. The advancement, postponement, and elimination of Hopf bifurcation are achieved without changing the system equilibrium point, and the mutation behavior of the transportation system is controlled so as to alleviate the traffic congestion. The changes in the stability of complex traffic systems are explained through the bifurcation analysis, which can better capture the characteristics of the traffic flow. By adjusting the control parameters in the feedback controllers, the influence of the boundary conditions on the stability of the traffic system is adequately described, and the effects of the unstable focuses and saddle points on the system are suppressed to slow down the traffic flow. In addition, the unstable bifurcation points can be eliminated and the Hopf bifurcation can be controlled to advance, delay, and disappear,so as to realize the control of the stability behavior of the traffic system, which can help to alleviate the traffic congestion and describe the actual traffic phenomena as well.

Bifurcation and Turing Pattern Formation in a Diffusion Modified Leslie-Gower Predator-Prey Model with Crowley-Martin Functional Response

In this paper, we study a modified Leslie-Gower predator-prey model with Smith growth subject to homogeneous Neumann boundary condition, in which the functional response is the Crowley-Martin functional response term. Firstly, for ODE model, the local stability of equilibrium point is given. And by using bifurcation theory and selecting suitable bifurcation parameters, we find many kinds of bifurcation phenomena, including Transcritical bifurcation and Hopf bifurcation. For the reaction-diffusion model, we find that Turing instability occurs. Besides, it is proved that Hopf bifurcation exists in the model. Finally, numerical simulations are presented to verify and illustrate the theoretical results.

Hopf bifurcation analysis and circuit implementation for a novel four-wing hyper-chaotic system

In the paper, a novel four-wing hyper-chaotic system is proposed and analyzed. A rare dynamic phenomenon is found that this new system with one equilibrium generates a four-wing-hyper-chaotic attractor as parameter varies. The system has rich and complex dynamical behaviors, and it is investigated in terms of Lyapunov exponents, bifurcation diagrams, Poincare maps, frequency spectrum, and numerical simulations. In addition, the theoretical analysis shows that the system undergoes a Hopf bifurcation as one parameter varies, which is illustrated by the numerical simulation. Finally, an analog circuit is designed to implement this hyper-chaotic system.

HOPF BIFURCATION OF AN OSCILLATOR WITH QUADRATIC AND CUBIC NONLINEARITIES AND WITH DELAYED VELOCITY FEEDBACK

This paper studies the local dynamics of an SDOF system with quadratic and cubic stiffness terms,and with linear delayed velocity feedback.The analysis indicates that for a sufficiently large velocity feedback gain,the equilibrium of the system may undergo a number of stability switches with an increase of time delay,and then becomes unstable forever.At each critical value of time delay for which the system changes its stability,a generic Hopf bifurcation occurs and a periodic motion emerges in a one-sided neighbourhood of the critical time delay.The method of Fredholm alternative is applied to determine the bifurcating periodic motions and their stability.It stresses on the effect of the system parameters on the stable regions and the amplitudes of the bifurcating periodic solutions.

HOPF BIFURCATION OF A NONLINEAR RESTRAINED CURVED PIPE CONVEYING FLUID BY DIFFERENTIAL QUADRATURE METHOD

This paper proposes a new method for investigating the Hopf bifurcation of a curved pipe conveying fluid with nonlinear spring support.The nonlinear equation of motion is derived by forces equilibrium on microelement of the system under consideration.The spatial coordinate of the system is discretized by the differential quadrature method and then the dynamic equation is solved by the Newton-Raphson method.The numerical solutions show that the inner fluid velocity of the Hopf bifurcation point of the curved pipe varies with different values of the parameter, nonlinear spring stiffness.Based on this,the cycle and divergent motions are both found to exist at specific fluid flow velocities with a given value of the nonlinear spring stiffness.The results are useful for further study of the nonlinear dynamic mechanism of the curved fluid conveying pipe.

A modified averaging scheme with application to the secondary Hopf bifurcation of a delayed van der Pol oscillator

In this paper, a modified averaging scheme is presented for a class of time-delayed vibration systems with slow variables. The new scheme is a combination of the averaging techniques proposed by Hale and by Lehman and Weibel, respectively. The averaged equation obtained from the modified scheme is simple enough but it retains the required information for the local nonlinear dynamics around an equilibrium. As an application of the present method, the delay value for which a secondary Hopf bifurcation occurs is successfully located for a delayed van der Pol oscillator.

Double Hopf bifurcation of composite laminated piezoelectric plate subjected to external and internal excitations

The double Hopf bifurcation of a composite laminated piezoelectric plate with combined external and internal excitations is studied. Using a multiple scale method, the average equations are obtained in two coordinates. The bifurcation response equations of the composite laminated piezoelectric plate with the primary parameter resonance, i.e., 1：3 internal resonance, are achieved. Then, the bifurcation feature of bifurcation equations is considered using the singularity theory. A bifurcation diagram is obtained on the parameter plane. Different steady state solutions of the average equations are analyzed. By numerical simulation, periodic vibration and quasi-periodic vibration responses of the Composite laminated piezoelectric plate are obtained.

Stability and Hopf bifurcation of a nonlinear electromechanical coupling system with time delay feedback

The stability and the Hopf bifurcation of a nonlinear electromechanical coupling system with time delay feedback are studied. By considering the energy in the air-gap field of the AC motor, the dynamical equation of the electromechanical coupling transmission system is deduced and a time delay feedback is introduced to control the dynamic behaviors of the system. The characteristic roots and the stable regions of time delay are determined by the direct method, and the relationship between the feedback gain and the length summation of stable regions is analyzed. Choosing the time delay as a bifurcation parameter, we find that the Hopf bifurcation occurs when the time delay passes through a critical value.A formula for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is given by using the normal form method and the center manifold theorem. Numerical simulations are also performed, which confirm the analytical results.

Diffusion-driven instability and Hopf bifurcation in Brusselator system

The Hopfbifurcation for the Brusselator ordinary-differential-equation （ODE） model and the corresponding partial-differential-equation （PDE） model are investigated by using the Hopf bifurcation theorem. The stability of the Hopf bifurcation periodic solution is discussed by applying the normal form theory and the center manifold theorem. When parameters satisfy some conditions, the spatial homogenous equilibrium solution and the spatial homogenous periodic solution become unstable. Our results show that if parameters are properly chosen, Hopf bifurcation does not occur for the ODE system, but occurs for the PDE system.

STOCHASTIC HOPF BIFURCATION IN QUASIINTEGRABLE-HAMILTONIAN SYSTEMS

A new procedure is developed to study the stochastic Hopf bifurcation in quasi- integrable-Hamiltonian systems under the Gaussian white noise excitation.Firstly,the singular bound- aries of the first-class and their asymptotic stable conditions in probability are given for the averaged Ito differential equations about all the sub-system's energy levels with respect to the stochastic aver- aging method.Secondly,the stochastic Hopf bifurcation for the coupled sub-systems are discussed by defining a suitable bounded torus region in the space of the energy levels and employing the theory of the torus region when the singular boundaries turn into the unstable ones.Lastly,a quasi-integrable- Hamiltonian system with two degrees of freedom is studied in detail to illustrate the above procedure. Moreover,simulations by the Monte-Carlo method are performed for the illustrative example to verify the proposed procedure.It is shown that the attenuation motions and the stochastic Hopf bifurcation of two oscillators and the stochastic Hopf bifurcation of a single oscillator may occur in the system for some system's parameters.Therefore,one can see that the numerical results are consistent with the theoretical predictions.

Hopf bifurcation control of a Pan-like chaotic system

This paper is concerned with the Hopf bifurcation control of a modified Pan-like chaotic system. Based on the Routh-Hurwtiz theory and high-dimensional Hopf bifurcation theory, the existence and stability of the Hopf bifurcation depending on selected values of the system parameters are studied. The region of the stability for the Hopf bifurcation is investigated.By the hybrid control method, a nonlinear controller is designed for changing the Hopf bifurcation point and expanding the range of the stability. Discussions show that with the change of parameters of the controller, the Hopf bifurcation emerges at an expected location with predicted properties and the range of the Hopf bifurcation stability is expanded. Finally,numerical simulation is provided to confirm the analytic results.

Supercritical as well as subcritical Hopf bifurcation in nonlinear flutter systems

The Hopf bifurcations of an airfoil flutter system with a cubic nonlinearity are investigated, with the flow speed as the bifurcation parameter. The center manifold theory and complex normal form method are Used to obtain the bifurcation equation. Interestingly, for a certain linear pitching stiffness the Hopf bifurcation is both supercritical and subcritical. It is found, mathematically, this is caused by the fact that one coefficient in the bifurcation equation does not contain the first power of the bifurcation parameter. The solutions of the bifurcation equation are validated by the equivalent linearization method and incremental harmonic balance method.

Hopf bifurcation and uncontrolled stochastic trafficinduced chaos in an RED-AQM congestion control system

We study the Hopf bifurcation and the chaos phenomena in a random early detection-based active queue man- agement （RED-AQM） congestion control system with a communication delay. We prove that there is a critical value of the communication delay for the stability of the RED-AQM control system. Furthermore, we show that the system will lose its stability and Hopf bifurcations will occur when the delay exceeds the critical value. When the delay is close to its critical value, we demonstrate that typical chaos patterns may be induced by the uncontrolled stochastic traffic in the RED-AQM control system even if the system is still stable, which reveals a new route to the chaos besides the bifurcation in the network congestion control system. Numerical simulations are given to illustrate the theoretical results.

Stability Analysis and Hopf Bifurcation for ODE System of Predator-Prey Model with Mutual Interference

In this paper, the temporal and spatial patterns of a diffusive predator-prey model with mutual interference under homogeneous Neumann boundary conditions were studied. Specifically, first, taking the intrinsic growth rate of the predator as the parameter, we give a computational and theoretical analysis of Hopf bifurcation on the positive equilibrium for the ODE system. As well, we have discussed the conditions for determining the bifurcation direction and the stability of the bifurcating periodic solutions.

Hopf bifurcation analysis of Chen circuit with direct time delay feedback

Direct time delay feedback can make non-chaotic Chen circuit chaotic. The chaotic Chen circuit with direct time delay feedback possesses rich and complex dynamical behaviours. To reach a deep and clear understanding of the dynamics of such circuits described by delay differential equations, Hopf bifurcation in the circuit is analysed using the Hopf bifurcation theory and the central manifold theorem in this paper. Bifurcation points and bifurcation directions are derived in detail, which prove to be consistent with the previous bifurcation diagram. Numerical simulations and experimental results are given to verify the theoretical analysis. Hopf bifurcation analysis can explain and predict the periodical orbit （oscillation） in Chen circuit with direct time delay feedback. Bifurcation boundaries are derived using the Hopf bifurcation analysis, which will be helpful for determining the parameters in the stabilisation of the originally chaotic circuit.