In this paper, a novel four dimensional hyper-chaotic system is coined based on the Chen system, which contains two quadratic terms and five system parameters. The proposed system can generate a hyper-chaotic attracto...In this paper, a novel four dimensional hyper-chaotic system is coined based on the Chen system, which contains two quadratic terms and five system parameters. The proposed system can generate a hyper-chaotic attractor in wide parameters regions. By using the center manifold theorem and the local bifurcation theory, a pitchfork bifurcation is demonstrated to arise at the zero equilibrium point. Numerical analysis demonstrates that the hyper-cha^tic system can generate complex dynamical behaviors, e.g., a direct transition from quasi-periodic behavior to hyper-chaotic behavior. Finally, an electronic circuit is designed to implement the hyper-chaotic system, the experimental results are consist with the numerical simulations, which verifies the existence of the hyper-chaotic attractor. Due to the complex dynamic behaviors, this new hyper-cha^tic system is useful in the secure communication.展开更多
From the point of view of dynamics, the phenomenon of mode jumping in the imperfect pitchfork problem is discussed. The dynamical mechanism of model jumping of structures, such as plate and shell, that is brought abou...From the point of view of dynamics, the phenomenon of mode jumping in the imperfect pitchfork problem is discussed. The dynamical mechanism of model jumping of structures, such as plate and shell, that is brought about by the extremum instability, is explained. Finally, we give numerical simulation to show the validity of our results.展开更多
The clustering behavior of a mono-disperse granular gas is experimentally studied in an asymmetric two-compartment setup. Unlike the random clustering in either compartment in the case of symmetric configuration when ...The clustering behavior of a mono-disperse granular gas is experimentally studied in an asymmetric two-compartment setup. Unlike the random clustering in either compartment in the case of symmetric configuration when lowering the shaking strength to below a critical value, the directed clustering is observed, which corresponds to an imperfect pitchfork bifurcation. Numerical solutions of the flux equation using a modified simple flux function show qualitative agreements with the experimental results. The potential application of this asymmetric structure is discussed.展开更多
Based on the empirical rate law and the kinetic data reported in the literature, it is predicted that the iodate-arsenous acid reaction may exhibit pitchfork bifurcation phenomenon in CSTR with two inflows of the reac...Based on the empirical rate law and the kinetic data reported in the literature, it is predicted that the iodate-arsenous acid reaction may exhibit pitchfork bifurcation phenomenon in CSTR with two inflows of the reactants.展开更多
This study explores the nonlinear dynamics of a quasi-zero stiffness(QZS)vibration isolator coupled with a piezoelectric energy harvester connected to an RL-resonant circuit.The model of the system is formulated with ...This study explores the nonlinear dynamics of a quasi-zero stiffness(QZS)vibration isolator coupled with a piezoelectric energy harvester connected to an RL-resonant circuit.The model of the system is formulated with the Lagrangian mechanics,representing a two-degree-of-freedom nonlinear electromechanical system subject to harmonic base excitation under a 1:1 internal resonance condition.The model is normalized,and the conditions dictating monostable and bistable oscillation modes are identified.The bifurcation characteristics of the coupled system are analyzed in both oscillation modes by means of harmonic balance and continuation methods.The vibration isolation performance,with and without the coupled harvester,is evaluated in terms of displacement transmissibility to assess its dual functionalities for vibration isolation and energy harvesting.Analytical results demonstrate that integrating a piezoelectric harvester into a monostable QZS isolator under 1:1 internal resonance does not compromise its vibration isolation capability while enabling efficient energy harvesting at extremely low-frequency base excitation.Furthermore,the system's response under strong base excitation is investigated exclusively for energy harvesting in both monostable and bistable modes,leading to optimal structural parameter design.The conditions for intra-well and inter-well periodic oscillation modes,as well as chaotic responses,are analyzed analytically and validated numerically through stability charts,basins of attraction,bifurcation diagrams,time histories,and Poincarémaps.This work provides a comprehensive understanding of the oscillation dynamics of QZS isolators and offers valuable insights for optimizing their geometric parameters to function as high-performance vibration isolators and/or energy harvesters.展开更多
In this paper. the classification of the pitchfork bifurcation in nonlinear problems withseveral parameters is considered. The regular extended systems for the pitchfork bifurcationpoints with different singularity ar...In this paper. the classification of the pitchfork bifurcation in nonlinear problems withseveral parameters is considered. The regular extended systems for the pitchfork bifurcationpoints with different singularity are proposed. An efficient algorithm for solving the ex-tended systems is given. Finally, some numerical examples are shown to test the efficiencyof this algorithm.展开更多
This paper is concerned with the computation of Hopf branches emanating from a Hopf/Pitchfork point in a two-parametor nonlinear problem satisfying a Z2symnletry condition. Our aim is to present a new al,proach to ...This paper is concerned with the computation of Hopf branches emanating from a Hopf/Pitchfork point in a two-parametor nonlinear problem satisfying a Z2symnletry condition. Our aim is to present a new al,proach to the theoretical and computational analysis of the bifurcating Hopf branches at this singular point by using the system designed to calculate Hopf points and exploring its symmetry. It is shown that a Hopf/Pitchfork point is a pitchfork bifurcation point in the system.Hence standard continuation and branch-switching can be used to compute these Hopf branches. In addition, an effect method based on the extended system of the singular points is developed for the computation of branch of secondary (nonsymmetric) Hopf points. The implementation of Newton's method for solving the extended system is also discussed. A numerical example is given.Keyworks: Hopf/Pitchfork point, Z2-symmetry, Hopf point, bifurcation, Extended system展开更多
Provides information on a study which proposed a continuation method for switching solution branches at a pitchfork point. Problem of the standard pitchfork; Information on the systems of equations; Numerical examples...Provides information on a study which proposed a continuation method for switching solution branches at a pitchfork point. Problem of the standard pitchfork; Information on the systems of equations; Numerical examples of the efficiency of the proposed switching method.展开更多
In this paper,we present a criterion for pitchfork bifurcations of smooth vector elds based on a topological argument.Our result expands Rajapakse and Smale's result[15]signi cantly.Based on our criterion,we prese...In this paper,we present a criterion for pitchfork bifurcations of smooth vector elds based on a topological argument.Our result expands Rajapakse and Smale's result[15]signi cantly.Based on our criterion,we present a class of families of non-symmetric vector elds undergoing a pitchfork bifurcation.展开更多
基金Project supported by the Natural Science Foundation of China (Grant Nos.61174094, 50977063, and 60904063)the Foundation of the Application Base and Frontier Technology Research Project of Tianjin, China (Grant No.10JCZDJC23100)the Development of Science and Technology Foundation of the Higher Education Institutions of Tianjin, China (Grant No.20080826)
文摘In this paper, a novel four dimensional hyper-chaotic system is coined based on the Chen system, which contains two quadratic terms and five system parameters. The proposed system can generate a hyper-chaotic attractor in wide parameters regions. By using the center manifold theorem and the local bifurcation theory, a pitchfork bifurcation is demonstrated to arise at the zero equilibrium point. Numerical analysis demonstrates that the hyper-cha^tic system can generate complex dynamical behaviors, e.g., a direct transition from quasi-periodic behavior to hyper-chaotic behavior. Finally, an electronic circuit is designed to implement the hyper-chaotic system, the experimental results are consist with the numerical simulations, which verifies the existence of the hyper-chaotic attractor. Due to the complex dynamic behaviors, this new hyper-cha^tic system is useful in the secure communication.
基金This work is supported by the Foundation of the National Educational Committeeand the National Natural Sciences Foundation of China.
文摘From the point of view of dynamics, the phenomenon of mode jumping in the imperfect pitchfork problem is discussed. The dynamical mechanism of model jumping of structures, such as plate and shell, that is brought about by the extremum instability, is explained. Finally, we give numerical simulation to show the validity of our results.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11034010 and 11274354)the Chinese Academy of Sciences "Strategic Priority Research Program - SJ-10" (Grant No. XDA04020200)the Special Fund for Earthquake Research of China (Grant No. 201208011)
文摘The clustering behavior of a mono-disperse granular gas is experimentally studied in an asymmetric two-compartment setup. Unlike the random clustering in either compartment in the case of symmetric configuration when lowering the shaking strength to below a critical value, the directed clustering is observed, which corresponds to an imperfect pitchfork bifurcation. Numerical solutions of the flux equation using a modified simple flux function show qualitative agreements with the experimental results. The potential application of this asymmetric structure is discussed.
文摘Based on the empirical rate law and the kinetic data reported in the literature, it is predicted that the iodate-arsenous acid reaction may exhibit pitchfork bifurcation phenomenon in CSTR with two inflows of the reactants.
基金Project supported by the National Key R&D Program of China(No.2023YFE0125900)。
文摘This study explores the nonlinear dynamics of a quasi-zero stiffness(QZS)vibration isolator coupled with a piezoelectric energy harvester connected to an RL-resonant circuit.The model of the system is formulated with the Lagrangian mechanics,representing a two-degree-of-freedom nonlinear electromechanical system subject to harmonic base excitation under a 1:1 internal resonance condition.The model is normalized,and the conditions dictating monostable and bistable oscillation modes are identified.The bifurcation characteristics of the coupled system are analyzed in both oscillation modes by means of harmonic balance and continuation methods.The vibration isolation performance,with and without the coupled harvester,is evaluated in terms of displacement transmissibility to assess its dual functionalities for vibration isolation and energy harvesting.Analytical results demonstrate that integrating a piezoelectric harvester into a monostable QZS isolator under 1:1 internal resonance does not compromise its vibration isolation capability while enabling efficient energy harvesting at extremely low-frequency base excitation.Furthermore,the system's response under strong base excitation is investigated exclusively for energy harvesting in both monostable and bistable modes,leading to optimal structural parameter design.The conditions for intra-well and inter-well periodic oscillation modes,as well as chaotic responses,are analyzed analytically and validated numerically through stability charts,basins of attraction,bifurcation diagrams,time histories,and Poincarémaps.This work provides a comprehensive understanding of the oscillation dynamics of QZS isolators and offers valuable insights for optimizing their geometric parameters to function as high-performance vibration isolators and/or energy harvesters.
基金广州市科技计划(批准号:201707010426和20180401350)广东省自然科学基金(批准号:2017A030313010)+3 种基金the Ministry of Economy and Competitiveness(批准号:MTM 2016-77278-P)Agencia de Gestio d’Ajuts Universitaris i de Recerca(批准号:2017SGR1617)the European project(批准号:Dynamics-H2020-MSCA-RISE-2017-777911)Barcelona Graduate School of Mathematics(批准号:MDM-2014-0445)资助项目
文摘In this paper. the classification of the pitchfork bifurcation in nonlinear problems withseveral parameters is considered. The regular extended systems for the pitchfork bifurcationpoints with different singularity are proposed. An efficient algorithm for solving the ex-tended systems is given. Finally, some numerical examples are shown to test the efficiencyof this algorithm.
文摘This paper is concerned with the computation of Hopf branches emanating from a Hopf/Pitchfork point in a two-parametor nonlinear problem satisfying a Z2symnletry condition. Our aim is to present a new al,proach to the theoretical and computational analysis of the bifurcating Hopf branches at this singular point by using the system designed to calculate Hopf points and exploring its symmetry. It is shown that a Hopf/Pitchfork point is a pitchfork bifurcation point in the system.Hence standard continuation and branch-switching can be used to compute these Hopf branches. In addition, an effect method based on the extended system of the singular points is developed for the computation of branch of secondary (nonsymmetric) Hopf points. The implementation of Newton's method for solving the extended system is also discussed. A numerical example is given.Keyworks: Hopf/Pitchfork point, Z2-symmetry, Hopf point, bifurcation, Extended system
基金NNSF of China, the Youth Science Foundation of Shanghai Higher Education(99QA80) and Science Foundation of Shanghai Science an
文摘Provides information on a study which proposed a continuation method for switching solution branches at a pitchfork point. Problem of the standard pitchfork; Information on the systems of equations; Numerical examples of the efficiency of the proposed switching method.
基金The second author was supported by the Smale Institute.This work was finished during the third author's stay in Graduate Center of City University of New York.
文摘In this paper,we present a criterion for pitchfork bifurcations of smooth vector elds based on a topological argument.Our result expands Rajapakse and Smale's result[15]signi cantly.Based on our criterion,we present a class of families of non-symmetric vector elds undergoing a pitchfork bifurcation.
