Studies On Complicated Dynamics Of Six-dimensional Non-autonomous Nonlinear Systems And Applications

With the development of science and technology, study of the dynamical problem forthe low-dimensional nonlinear dynamical system cannot satisfy the requirement in someof practical engineering. In order to accurately analyze the complicated dynamicalproblem of nonlinear system, the higher-dimensional nonlinear dynamical system needs tobe considered. However, compared with the research of the dynamical problem for thelow-dimensional nonlinear system, it is difficult to study the theory of high-dimensionalnonlinear system. Especially, In order to obtain the theoretical results which closer to theoriginal system, the complicated dynamics of the high-dimensional non-autonomousnonlinear system is investigated directly. The process of simplification is reducedcomparing with the previous methods. Then more characteristics of the original system areretained. Therefore, study of the complicated dynamics for the high-dimensionalnon-autonomous nonlinear system is very important subject in science and engineeringapplications.Based on the method of study four-dimensional non-autonomous nonlineardynamical system, some of methods are improved and applied to study thesix-dimensional non-autonomous nonlinear dynamical system. The complicated dynamicsof some six-dimensional non-autonomous nonlinear systems is investigated using thesemethods as follows.(1) The multi-degree-of-freedom Melnikov method which studied the single-pulsechaotic dynamics is improved to investigate the six-dimensional non-autonomousnonlinear dynamical system. The single-pulse chaotic dynamics of a four-edge simplysupported buckled rectangular thin plate under in-plane excitation are investigated usingthe improved method. Firstly, the Galerkin method is employed to discretize the motionequations of the four-edge simply supported buckled rectangular thin plate under in-planeexcitation. A non-autonomous nonlinear dynamics equation with three-degree-of-freedomis derived. Then, the three-order normal form is used to analyze this system. Somenonlinear terms in this system have less effect than other terms. So these terms areconsidered as perturbation terms. In the end, the single-pulse chaotic motions of thefour-edge simply supported buckled rectangular thin plate under in-plane excitation arefound from theoretical analysis and numerical simulation. Furthermore, the single-pulsechaotic dynamics of a four-edge simply supported buckled rectangular thin plate under thecombination of in-plane and transversal excitations is investigated using this method.(2) The extended Melnikov method which studied the multi-pulse chaotic dynamicsis improved to investigate the six-dimensional non-autonomous nonlinear dynamicalsystem in mixed coordinate. In the process of investigation, when a cross-section is appled to the six-dimensional non-autonomous nonlinear dynamical system, a seven-dimensionalautonomous nonlinear dynamical system is obtained. The finite inductive approach is usedto prove the Melnikov function.The multi-pulse chaotic dynamics of three kinds of plate are investigated using theimproved extended Melnikov method. Such as the four-edge simply supported buckledrectangular thin plate under in-plane excitation, the four-edge simply supported buckledrectangular thin plate under the combination of in-plane and transversal excitations and afour-edge simply supported composite laminated piezoelectric rectangular plate underin-plane and transversal excitations. In the process of computation, because these systemsare too complex to study using the improved method, the three-order normal form is usedto simplify the three-degree-of-freedom nonlinear dynamical equations. Then the lastfour-dimensional equations are transformed into the polar form, and the first two equationsare decoupled with the last four equations by coordinate transformation. A homoclinicbifurcation is found from the first two equations and a multi-pulse Shilnikov type chaoticmotion is found from the last four equations when studing these decoupled systems. Whencalculating the Melnikov functions, some of integrals are too complex to calculate.The k-pulse Melnikov function of these systems are obtained using the Taylor series andthe residue theory. In the end, the multi-pulse chaotic motions of these systems are foundfrom the numerical simulations which further verify the result of theoretical analysis.(3) The extended Melnikov method is improved to investigate the six-dimensionalnon-autonomous nonlinear dynamical system in Cartesian coordinate and correctness ofsuch a method is theoretically proved. In the process of investigation, when a cross-sectionis appled to six-dimensional non-autonomous nonlinear dynamical system, aseven-dimensional autonomous nonlinear dynamical system is obtained.The multi-pulse chaotic dynamics of a four-edge simply supported compositelaminated rectangular plate subjected to transversal excitation and a four-edge simplysupported honeycomb sandwich rectangular plate under in-plane and transversalexcitations are investigated using the improved extended Melnikov method. In the processof computation, the three-order normal form is used to analyze these systems. Somenonlinear terms of these systems disappear after the normal form calculation. By virtue ofthe theory of the normal form, these nonlinear terms in these systems have less effect thanother terms. So these terms are considered as perturbation terms in these systems. Theunperturbed systems of these systems are transformed into Hamiltonian systems and thefirst two equations are decoupled with the last four equations by coordinatetransformation. When studing the first two equations of these decoupled systems, aheteroclinic orbit is found from the four-edge simply supported laminated rectangularplate subjected to transversal excitation, and a homoclinic orbit is found from thefour-edge simply supported honeycomb sandwich rectangular plate under in-plane and transversal excitations. Multi-pulse Shilnikov type chaotic motions of these systems arefound in the condition of resonance. In the end, multi-pulse chaotic motions of thesesystems are found from numerical simulations which further verify the result of theoreticalanalysis.