Padé Approximation And The Complicated Dynamics Of Nonlinear Dynamical System

Discovery of complex phenomena and establishment of bifurcation and chaostheory in nonlinear dynamics is one of the major achievements of the contemporaryfield of basic science, which inspired many scholars depth study and exploration. Padé approximation has been used effectively to solve the zeros of the function,numerical integration, analytical solution of ordinary differential equations, numericalsolution of partial differential equations and other fields in the field of numericalanalysis. This is the further promotion of this approach in field of complex problemsof nonlinear dynamical systems for:①studying the dynamic behavior in strongnonlinear vibration system with non-Z2symmetric;②improving the solving accuracyof homoclinic and heteroclinic bifurcation problem based on Melnikov method innon-Z2symmetric systems；③obtaining universal analytical method for homoclinicorbits and precise chaotic threshold value in three-dimensional system. Combine withthe theory of nonlinear dynamics to carry out research work and propose effectivesolutions.The main content and results of research in this paper are reflected in thefollowing aspects:(1). Based on Padé approximation resolution methods, which are given tocalculate the problems of homoclinic and heteroclinic bifurcation in two-well non-Z2symmetric systems under the action of strength and weak disturbance, can overcomelimitations of the application of analysis method in weakly nonlinear Z2symmetrysystems in this field.(2). The Padé approximation is used to propose the simple way which improvethe calculated accuracy of chaotic threshold when Melnikov function is used toanalysis nonlinear dynamical systems with non-Z2symmetry. For non-Z2symmetricthree-well potential energy systems under the action of complex forms of incentives,through directly put disturbance into the process of calculation to reflect affection ofthe non-Z2symmetry and higher order nonlinear part in the Melnikov functionexpression, and then the critical values of chaos are obtained from two angles ofhomoclinic and heteroclinic bifurcation. (3). The problem of complex heteroclinic orbit in non-Z2symmetric system isstudied. Based on the non-Z2symmetry the complex heteroclinic orbit is divided intothree categories, which are discussed in detail from causes and characteristics. Thesolution form which meet nature of complex heteroclinic orbit is proposed, andcombine with Padé approximation and convergence condition which is heteroclinicorbit tend to asymmetric saddle point respectively, then the analytical expressions ofthese three categories heteroclinic orbit are obtained.(4). The problem of complex homoclinic bifurcation in multi-well non-Z2symmetric system is studied. According to the energy function discussion and Padéapproximation method to study the special homoclinic orbit with non-Z2symmetrydue to presence of non-Z2symmetry and higher order nonlinear part in multi-wellpotential energy system, to establish the corresponding relationship between thesystem parameters and the position of equilibrium point at this time, and to obtainprecise value of homoclinic bifurcation parameters by means of the Melnikov method.(5). The model of heterogeneous thin elastic rod is established to study process ofcombining the DNA and protein molecules, and critical buckling behavior of thinelastic rod is study through analytical way. Mathematical model of thin elastic rodstatic configuration is established with arc length as independent variable by appliedCosserat medium theory, and the fractional differential equations with complexnonlinear terms are obtained. Complex dynamical behavior in this system is analyzed,spatial configuration of thin elastic rod, which is corresponding with heteroclinic orbitphenomenon, is discussed by introducing scale transformation of independent variableinto Padé approximation.(6). The Samle chaos in three-dimensional nonlinear dynamical system is studied.Three-dimensional dynamical system with more complex nonlinear terms as researchobject (the systems with Shilnikov and Lorenz type homoclinic orbits), using thecomputing ideas of Padé approximation method, and setting local analytical solutionnear initial value as a bridge between stable and unstable manifolds at equilibriumpoint, analytical method is established to obtained Samle horseshoes chaotic motionthreshold value directly and analytical expressions of homoclinic orbits.