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...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.展开更多
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 demonstr...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.展开更多
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...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.展开更多
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 red...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.展开更多
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 b...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.展开更多
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,th...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.展开更多
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 stabil...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.展开更多
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 ...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.展开更多
This paper investigates the dynamics of a TCP system described by a first- order nonlinear delay differential equation. By analyzing the associated characteristic transcendental equation, it is shown that a Hopf bifur...This paper investigates the dynamics of a TCP system described by a first- order nonlinear delay differential equation. By analyzing the associated characteristic transcendental equation, it is shown that a Hopf bifurcation sequence occurs at the pos- itive equilibrium as the delay passes through a sequence of critical values. The explicit algorithms for determining the Hopf bifurcation direction and the stability of the bifur- cating periodic solutions are derived with the normal form theory and the center manifold theory. The global existence of periodic solutions is also established with the method of Wu (Wu, J. H. Symmetric functional differential equations and neural networks with memory. Transactions of the American Mathematical Society 350(12), 4799-4838 (1998)).展开更多
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 va...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.展开更多
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 ...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.展开更多
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...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.展开更多
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 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.展开更多
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 t...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.展开更多
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 a...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.展开更多
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 ob...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.展开更多
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 val...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.展开更多
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...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.展开更多
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 o...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.展开更多
The ecological model of a class of the two microbe populations with second-order growth rate was studied. The methods of qualitative theory of ordinary differential equations were used in the four-dimension phase spac...The ecological model of a class of the two microbe populations with second-order growth rate was studied. The methods of qualitative theory of ordinary differential equations were used in the four-dimension phase space. The qualitative property and stability of equilibrium points were analysed. The conditions under which the positive equilibrium point exists and becomes and O+ attractor are obtained. The problems on Hopf bifurcation are discussed in detail when small perturbation occurs.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant No. 62073172)the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20221329)。
文摘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.
文摘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.
基金the National Natural Science Foundations of China(Grant Nos.62171401 and 62071411).
文摘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.
基金This work was supported by the grants from the National Nat-ural Science Foundation of China(No.11772002)Ningxia higher education first-class discipline construction funding project(No.NXYLXK2017B09)+2 种基金Major Special project of North Minzu University(No.ZDZX201902)Open project of The Key Laboratory of In-telligent Information and Big Data Processing of NingXia Province(No.2019KLBD008)Postgraduate Innovation Project of North Minzu University(No.YCX22099).
文摘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.
文摘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.
基金The project supported by the National Natural Science Foundation of China (19972025)
文摘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.
基金Project supported by the National Natural Science Foundation of China (No.10771032)the Natural Science Foundation of Jiangsu Province (BK2006088)
文摘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.
基金Project supported by the National Natural Science Foundation of China(Grant No.11372102)
文摘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.
基金Project supported by the National Natural Science Foundation of China (Nos. 10771215 and10771094)
文摘This paper investigates the dynamics of a TCP system described by a first- order nonlinear delay differential equation. By analyzing the associated characteristic transcendental equation, it is shown that a Hopf bifurcation sequence occurs at the pos- itive equilibrium as the delay passes through a sequence of critical values. The explicit algorithms for determining the Hopf bifurcation direction and the stability of the bifur- cating periodic solutions are derived with the normal form theory and the center manifold theory. The global existence of periodic solutions is also established with the method of Wu (Wu, J. H. Symmetric functional differential equations and neural networks with memory. Transactions of the American Mathematical Society 350(12), 4799-4838 (1998)).
基金supported by the National Natural Science Foundation of China(Grant Nos.10772135 and 60874028)the Young Scientists Fund of the National Natural Science Foundation of China(Grant No.11202148)+2 种基金the Incentive Funding of the National Research Foundation of South Africa(GrantNo.IFR2009090800049)the Eskom Tertiary Education Support Programme of South Africathe Research Foundation of Tianjin University of Science and Technology
文摘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.
基金Project supported by the National Natural Science Foundation of China(No.10272051).
文摘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.
基金FANEDD of China (200430)the National Natural Science Foundation of China (10372116,10532050)
文摘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.
基金Project supported by the National Natural Science Foundation of China(Nos.11402127,11290152 and 11072008)
文摘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.
基金Project supported by the National Natural Science Foundation of China(Grant No.61104040)the Natural Science Foundation of Hebei Province,China(Grant No.E2012203090)the University Innovation Team of Hebei Province Leading Talent Cultivation Project,China(Grant No.LJRC013)
文摘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.
基金The project supported by the National Natural Science Foundation of China (10302025)
文摘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.
基金Project supported by the National Natural Science Foundation of China (No.10772202)the Doctoral Foundation of Ministry of Education of China (No.20050558032)the Natural Science Foundation of Guangdong Province (Nos.07003680 and 05003295)
文摘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.
基金Project supported by the National High Technology Research and Development Program of China (Grant No.2007AA01Z480)
文摘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.
文摘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.
基金Project supported in part by the National Natural Science Foundation of China (Grant No. 60804040)Fok Ying-Tong Education Foundation for Young Teacher (Grant No. 111065)
文摘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.
文摘The ecological model of a class of the two microbe populations with second-order growth rate was studied. The methods of qualitative theory of ordinary differential equations were used in the four-dimension phase space. The qualitative property and stability of equilibrium points were analysed. The conditions under which the positive equilibrium point exists and becomes and O+ attractor are obtained. The problems on Hopf bifurcation are discussed in detail when small perturbation occurs.