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 proposes a novel reconfigurable Goldberg 6R linkage,conformed to the construction of variant serial Goldberg 6R linkage,while simultaneously satisfying the line-symmetric Bricard qualifications.The isomeric...This paper proposes a novel reconfigurable Goldberg 6R linkage,conformed to the construction of variant serial Goldberg 6R linkage,while simultaneously satisfying the line-symmetric Bricard qualifications.The isomeric mechanism of this novel reconfigurable mechanism is obtained in combination with the isomerization method.The geometrically constrained conditions result in variable motion branches of the mechanism.Based on the singular value decomposition of the Jacobian matrix,the motion branches and branch bifurcation characteristics are analyzed,and the schematics of bifurcations in joint space is derived.This novel 6R linkage features one Goldberg 6R motion branch,two line-symmetric Bricard 6R motion branches,and one Bennett motion branch.With regards to the line-symmetric Bricard 6R motion branches,a similar function for the disassembly and recombination process can be achieved by reconstructing an intermediate configuration through bifurcation.Then,the isomerized generalized variant Goldberg 6R linkage is explicated in a similar way.Acting as a bridge,reconfigurability connects two families of overconstrained mechanisms.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
After Bénard＇s experiment in 1900, Rayleigh formulated heat convection problems by the Oberbeck-Boussinesq approximation in the horizontal strip domain in 1916. The pattern formations have been investigated by t...After Bénard＇s experiment in 1900, Rayleigh formulated heat convection problems by the Oberbeck-Boussinesq approximation in the horizontal strip domain in 1916. The pattern formations have been investigated by the bifurcation theory, weakly nonlinear theories and computational approaches. The boundary conditions for the velocity on the upper and lower boundaries are usually assumed as stress-free or no-slip. In the first part of this paper, some bifurcation pictures for the case of the stress-free on the upper boundary and the no-slip on the lower boundary are obtained. In the second part of this paper, the bifurcation pictures for the case of the stress-free on both boundaries by a computer assisted proof are verified. At last., Bénard-Marangoni heat convections for the ease of the free surface of the upper boundary are considered.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.
基金Projects(51535008,51721003)supported by the National Natural Science Foundation of ChinaProject(B16034)supported by the Program of Introducing Talents of Discipline to Universities(“111 Program”),China。
文摘This paper proposes a novel reconfigurable Goldberg 6R linkage,conformed to the construction of variant serial Goldberg 6R linkage,while simultaneously satisfying the line-symmetric Bricard qualifications.The isomeric mechanism of this novel reconfigurable mechanism is obtained in combination with the isomerization method.The geometrically constrained conditions result in variable motion branches of the mechanism.Based on the singular value decomposition of the Jacobian matrix,the motion branches and branch bifurcation characteristics are analyzed,and the schematics of bifurcations in joint space is derived.This novel 6R linkage features one Goldberg 6R motion branch,two line-symmetric Bricard 6R motion branches,and one Bennett motion branch.With regards to the line-symmetric Bricard 6R motion branches,a similar function for the disassembly and recombination process can be achieved by reconstructing an intermediate configuration through bifurcation.Then,the isomerized generalized variant Goldberg 6R linkage is explicated in a similar way.Acting as a bridge,reconfigurability connects two families of overconstrained mechanisms.
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.
文摘After Bénard＇s experiment in 1900, Rayleigh formulated heat convection problems by the Oberbeck-Boussinesq approximation in the horizontal strip domain in 1916. The pattern formations have been investigated by the bifurcation theory, weakly nonlinear theories and computational approaches. The boundary conditions for the velocity on the upper and lower boundaries are usually assumed as stress-free or no-slip. In the first part of this paper, some bifurcation pictures for the case of the stress-free on the upper boundary and the no-slip on the lower boundary are obtained. In the second part of this paper, the bifurcation pictures for the case of the stress-free on both boundaries by a computer assisted proof are verified. At last., Bénard-Marangoni heat convections for the ease of the free surface of the upper boundary are considered.
