Complex dynamics are studied in the T system, a three-dimensional autonomous nonlinear system. In particular, we perform an extended Hopf bifurcation analysis of the system. The periodic orbit immediately following th...Complex dynamics are studied in the T system, a three-dimensional autonomous nonlinear system. In particular, we perform an extended Hopf bifurcation analysis of the system. The periodic orbit immediately following the Hopf bifurcation is constructed analytically for the T system using the method of multiple scales, and the stability of such orbits is analyzed. Such analytical results complement the numerical results present in the literature. The analytical results in the post-bifurcation regime are verified and extended via numerical simulations, as well as by the use of standard power spectra, autocorrelation functions, and fractal dimensions diagnostics. We find that the T system exhibits interesting behaviors in many parameter regimes.展开更多
This paper presents the application of bifurcation method on the steady state three-phase load-flow Jacobian method to study the voltage stability of unbalanced distribution systems. The eigenvalue analysis is used to...This paper presents the application of bifurcation method on the steady state three-phase load-flow Jacobian method to study the voltage stability of unbalanced distribution systems. The eigenvalue analysis is used to study distribution system behavior under different operating conditions. Two-bus connected by asymmetrical line is used as the study system. The effects of both unbalance and extreme loading conditions are investigated. Also, the impact of distributed energy resources is studied. Different case studies and loading scenarios are presented to trace the eigenvalues of the Jacobian matrix. The results exhibit the existence of a new bifurcation point which may not be related to the voltage stability.展开更多
In this paper, analytical and numerical studies are carried out on the full annular rub motions of a nonlinear Jeffcott rotor. Transition sets of the synchronous full annular rub are given with the help of averaging m...In this paper, analytical and numerical studies are carried out on the full annular rub motions of a nonlinear Jeffcott rotor. Transition sets of the synchronous full annular rub are given with the help of averaging method and constraint bifurcation theory to discuss the effects of system parameters on jump phenomena. Routh-Hurwitz criteria are employed to analyze the stability of synchronous full annular rub solution and determine the boundaries of static and Hopf bifurcations. Finally, the response and onset condition of reverse dry whip are investigated numerically, and at the same time, the influences of rotor parameters and rotation speed on the characteristics of the rotor response are investigated.展开更多
Using a combination of analytical and numerical methods, the paper studies bifurcations and chaotic motions of a two-dimensional airfoil with cubic nonlinearity in incompressible flow. One type of critical points (cha...Using a combination of analytical and numerical methods, the paper studies bifurcations and chaotic motions of a two-dimensional airfoil with cubic nonlinearity in incompressible flow. One type of critical points (characterized by a negative eigenvalue, a simple zero eigenvalue and a pair of purely imaginary eigenvalues) for the bifurcation response equations is considered. With the aid of the normal form theory, the explicit expressions of the critical bifurcation lines leading to incipient and secondary bifurcations are obtained. The stability of the bifurcation solutions is also investigated. By using the undetermined coefficient method, the homoclinic orbit is found, and the uniform convergence of the homoclinic orbit series expansion is proved. It analytically demonstrates that there exists a homoclinic orbit joining the initial equilibrium point to itself, therefore Smale horseshoe chaos occurs for this system via Si'lnikov criterion. The system evolves into chaotic motion through period-doubling bifurcation, and is periodic again as the dimensionless airflow speed increases. Numerical simulations are also given, which confirm the analytical results.展开更多
A high-codimension homoclinic bifurcation is considered with one orbit flip and two inclination flips accompanied by resonant principal eigenvalues. A local active coordinate system in a small neighborhood of homoclin...A high-codimension homoclinic bifurcation is considered with one orbit flip and two inclination flips accompanied by resonant principal eigenvalues. A local active coordinate system in a small neighborhood of homoclinic orbit is introduced. By analysis of the bifurcation equation, the authors obtain the conditions when the original flip homoclinic orbit is kept or broken. The existence and the existence regions of several double periodic orbits and one triple periodic orbit bifurcations are proved. Moreover, the complicated homoclinic-doubling bifurcations are found and expressed approximately.展开更多
This paper investigates the Hopf bifurcation of a 4-dimensional hyperchaotic system withonly one equilibrium.A detailed set of conditions are derived,which guarantee the existence of theHopf bifurcation.Furthermore,th...This paper investigates the Hopf bifurcation of a 4-dimensional hyperchaotic system withonly one equilibrium.A detailed set of conditions are derived,which guarantee the existence of theHopf bifurcation.Furthermore,the standard normal form theory is applied to determine the directionand type of the Hopf bifurcation,and the approximate expressions of bifurcating periodic solutions andtheir periods.In addition,numerical simulations are used to justify theoretical results.展开更多
A new simple piecewise linear map of the plane is presented and analyzed, then a detailed study of its dynamical behaviour is described, along with some other dynamical phenomena, especially fixed points and their sta...A new simple piecewise linear map of the plane is presented and analyzed, then a detailed study of its dynamical behaviour is described, along with some other dynamical phenomena, especially fixed points and their stability, observation of a new chaotic attractors obtained via border collision bifurcation. An important resuk about coexisting chaotic attractors is also numerically studied and discussed.展开更多
文摘Complex dynamics are studied in the T system, a three-dimensional autonomous nonlinear system. In particular, we perform an extended Hopf bifurcation analysis of the system. The periodic orbit immediately following the Hopf bifurcation is constructed analytically for the T system using the method of multiple scales, and the stability of such orbits is analyzed. Such analytical results complement the numerical results present in the literature. The analytical results in the post-bifurcation regime are verified and extended via numerical simulations, as well as by the use of standard power spectra, autocorrelation functions, and fractal dimensions diagnostics. We find that the T system exhibits interesting behaviors in many parameter regimes.
文摘This paper presents the application of bifurcation method on the steady state three-phase load-flow Jacobian method to study the voltage stability of unbalanced distribution systems. The eigenvalue analysis is used to study distribution system behavior under different operating conditions. Two-bus connected by asymmetrical line is used as the study system. The effects of both unbalance and extreme loading conditions are investigated. Also, the impact of distributed energy resources is studied. Different case studies and loading scenarios are presented to trace the eigenvalues of the Jacobian matrix. The results exhibit the existence of a new bifurcation point which may not be related to the voltage stability.
基金supported by the National Natural Science Foundation of China (Grant No. 10632040)
文摘In this paper, analytical and numerical studies are carried out on the full annular rub motions of a nonlinear Jeffcott rotor. Transition sets of the synchronous full annular rub are given with the help of averaging method and constraint bifurcation theory to discuss the effects of system parameters on jump phenomena. Routh-Hurwitz criteria are employed to analyze the stability of synchronous full annular rub solution and determine the boundaries of static and Hopf bifurcations. Finally, the response and onset condition of reverse dry whip are investigated numerically, and at the same time, the influences of rotor parameters and rotation speed on the characteristics of the rotor response are investigated.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10972099, 10632040)China Postdoctoral Science Foundation (Grant No. 20090450765)the Natural Science Foundation of Tianjin, China (Grant No. 09JCZDJC26800)
文摘Using a combination of analytical and numerical methods, the paper studies bifurcations and chaotic motions of a two-dimensional airfoil with cubic nonlinearity in incompressible flow. One type of critical points (characterized by a negative eigenvalue, a simple zero eigenvalue and a pair of purely imaginary eigenvalues) for the bifurcation response equations is considered. With the aid of the normal form theory, the explicit expressions of the critical bifurcation lines leading to incipient and secondary bifurcations are obtained. The stability of the bifurcation solutions is also investigated. By using the undetermined coefficient method, the homoclinic orbit is found, and the uniform convergence of the homoclinic orbit series expansion is proved. It analytically demonstrates that there exists a homoclinic orbit joining the initial equilibrium point to itself, therefore Smale horseshoe chaos occurs for this system via Si'lnikov criterion. The system evolves into chaotic motion through period-doubling bifurcation, and is periodic again as the dimensionless airflow speed increases. Numerical simulations are also given, which confirm the analytical results.
基金supported by the National Natural Science Foundation of China(No.11126097)
文摘A high-codimension homoclinic bifurcation is considered with one orbit flip and two inclination flips accompanied by resonant principal eigenvalues. A local active coordinate system in a small neighborhood of homoclinic orbit is introduced. By analysis of the bifurcation equation, the authors obtain the conditions when the original flip homoclinic orbit is kept or broken. The existence and the existence regions of several double periodic orbits and one triple periodic orbit bifurcations are proved. Moreover, the complicated homoclinic-doubling bifurcations are found and expressed approximately.
基金supported by the National Natural Science Foundation of China under Grant Nos. 10871074, 10671105the Natural Science Foundation of Guangdong Province of China under Grant No. 05300162
文摘This paper investigates the Hopf bifurcation of a 4-dimensional hyperchaotic system withonly one equilibrium.A detailed set of conditions are derived,which guarantee the existence of theHopf bifurcation.Furthermore,the standard normal form theory is applied to determine the directionand type of the Hopf bifurcation,and the approximate expressions of bifurcating periodic solutions andtheir periods.In addition,numerical simulations are used to justify theoretical results.
文摘A new simple piecewise linear map of the plane is presented and analyzed, then a detailed study of its dynamical behaviour is described, along with some other dynamical phenomena, especially fixed points and their stability, observation of a new chaotic attractors obtained via border collision bifurcation. An important resuk about coexisting chaotic attractors is also numerically studied and discussed.
