This paper first analyzes the features of two classes of numerical methods for global analysis of nonlinear dynamical systems, which regard state space respectively as continuous and discrete ones. On basis of this un...This paper first analyzes the features of two classes of numerical methods for global analysis of nonlinear dynamical systems, which regard state space respectively as continuous and discrete ones. On basis of this understanding it then points out that the previously proposed method of point mapping under cell reference (PMUCR), has laid a frame work for the development of a two scaled numerical method suitable for the global analysis of high dimensional nonlinear systems, which may take the advantages of both classes of single scaled methods but will release the difficulties induced by the disadvantages of them. The basic ideas and main steps of implementation of the two scaled method, namely extended PMUCR, are elaborated. Finally, two examples are presented to demonstrate the capabilities of the proposed method.展开更多
In some mechanical nonlinear systems, the transient motion will be undergoing a very long process and the attractor-basin boundaries are so complicated that some difficulties occur in analyzing the system global behav...In some mechanical nonlinear systems, the transient motion will be undergoing a very long process and the attractor-basin boundaries are so complicated that some difficulties occur in analyzing the system global behavior. To solve this problem a mixed cell mapping method based on the point mapping and the principle of simple cell mapping is developed. The algorithm of the mixed cell mapping is studied. A dynamic model of a gear pair is established with the backlash, damping, transmission error and the time-varying stiffness taken into consideration. The global behaviors of this system are analyzed. The coexistence of the system attractors and the respective attractor-basin of each attractor with different parameters are obtained, thus laying a theoretical basis for improvement of the dynamic behaviors of gear system.展开更多
Three main parts of generalized cell mapping are improved for global analysis. A simple method, which is not based on the theory of digraphs, is presented to locate complete self-cycling sets that corre- spond to attr...Three main parts of generalized cell mapping are improved for global analysis. A simple method, which is not based on the theory of digraphs, is presented to locate complete self-cycling sets that corre- spond to attractors and unstable invariant sets involving saddle, unstable periodic orbit and chaotic saddle. Refinement for complete self-cycling sets is developed to locate attractors and unstable in- variant sets with high degree of accuracy, which can start with a coarse cell structure. A nonuniformly interior-and-boundary sampling technique is used to make the refinement robust. For homeomorphic dissipative dynamical systems, a controlled boundary sampling technique is presented to make gen- eralized cell mapping method with refinement extremely accurate to obtain invariant sets. Recursive laws of group absorption probability and expected absorption time are introduced into generalized cell mapping, and then an optimal order for quantitative analysis of transient cells is established, which leads to the minimal computational work. The improved method is applied to four examples to show its effectiveness in global analysis of dynamical systems.展开更多
The piezomagnetoelastic energy harvester system subjected to harmonic and Poisson white noise excitations is studied by using the generalized cell mapping method. The transient and stationary probability density funct...The piezomagnetoelastic energy harvester system subjected to harmonic and Poisson white noise excitations is studied by using the generalized cell mapping method. The transient and stationary probability density functions(PDFs) of response based on the global viewpoint are obtained by the matrix analysis method. Monte Carlo simulation results verify the accuracy of this method. It can be observed that evolutionary direction of transient and stationary PDFs is in accordance with the unstable manifold for this system, and a stochastic P-bifurcation occurs as the intensity of Poisson white noise increases. This study presents an efficient numerical tool to solve the stochastic response of a three-dimensional dynamical system and provides a new idea to analyze the energy harvester system.展开更多
基金supported by the National Natural Science Foundation of China (NSFC) (10872155)
文摘This paper first analyzes the features of two classes of numerical methods for global analysis of nonlinear dynamical systems, which regard state space respectively as continuous and discrete ones. On basis of this understanding it then points out that the previously proposed method of point mapping under cell reference (PMUCR), has laid a frame work for the development of a two scaled numerical method suitable for the global analysis of high dimensional nonlinear systems, which may take the advantages of both classes of single scaled methods but will release the difficulties induced by the disadvantages of them. The basic ideas and main steps of implementation of the two scaled method, namely extended PMUCR, are elaborated. Finally, two examples are presented to demonstrate the capabilities of the proposed method.
基金This work was supported by the National Natural Science Foundation of China (Grant No.50075070).
文摘In some mechanical nonlinear systems, the transient motion will be undergoing a very long process and the attractor-basin boundaries are so complicated that some difficulties occur in analyzing the system global behavior. To solve this problem a mixed cell mapping method based on the point mapping and the principle of simple cell mapping is developed. The algorithm of the mixed cell mapping is studied. A dynamic model of a gear pair is established with the backlash, damping, transmission error and the time-varying stiffness taken into consideration. The global behaviors of this system are analyzed. The coexistence of the system attractors and the respective attractor-basin of each attractor with different parameters are obtained, thus laying a theoretical basis for improvement of the dynamic behaviors of gear system.
基金Supported by the National Natural Science Foundation of China (Grant Nos. 10432010, 10502039)
文摘Three main parts of generalized cell mapping are improved for global analysis. A simple method, which is not based on the theory of digraphs, is presented to locate complete self-cycling sets that corre- spond to attractors and unstable invariant sets involving saddle, unstable periodic orbit and chaotic saddle. Refinement for complete self-cycling sets is developed to locate attractors and unstable in- variant sets with high degree of accuracy, which can start with a coarse cell structure. A nonuniformly interior-and-boundary sampling technique is used to make the refinement robust. For homeomorphic dissipative dynamical systems, a controlled boundary sampling technique is presented to make gen- eralized cell mapping method with refinement extremely accurate to obtain invariant sets. Recursive laws of group absorption probability and expected absorption time are introduced into generalized cell mapping, and then an optimal order for quantitative analysis of transient cells is established, which leads to the minimal computational work. The improved method is applied to four examples to show its effectiveness in global analysis of dynamical systems.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11302170,11202160,11302171the Fundamental Research Funds for the Central Universities under Grant No.3102014JCQ01079
文摘The piezomagnetoelastic energy harvester system subjected to harmonic and Poisson white noise excitations is studied by using the generalized cell mapping method. The transient and stationary probability density functions(PDFs) of response based on the global viewpoint are obtained by the matrix analysis method. Monte Carlo simulation results verify the accuracy of this method. It can be observed that evolutionary direction of transient and stationary PDFs is in accordance with the unstable manifold for this system, and a stochastic P-bifurcation occurs as the intensity of Poisson white noise increases. This study presents an efficient numerical tool to solve the stochastic response of a three-dimensional dynamical system and provides a new idea to analyze the energy harvester system.
基金supported by grants (11172197,11332008,and 11572215) from the National Natural Science Foundation of Chinaa grant from the University of California Institute for Mexico and the United States (UC MEXUS)the Consejo Nacional de Cienciay Tecnologia de Mexico (CONACYT) through the project "Hybridizing Set Oriented Methods and Evolutionary Strategies to Obtain Fast and Reliable Multi-objective Optimization Algorithms"
