Solution Of Bifurcation In Nonlinear Block Equation

After Block equations which described nuclear magnetic resonance(NMR) were found in 1946, they were modified by adding many nonlinear terms for different demands. For example, nonlinear feedback terms were added into the equations in order to explain phenomena of magnetization-dependent and nonlinear Block equations were obtained. Then dynamics such as bifurcation and chaos of these modified equations were given more attention. This dissertation mainly presents a study on stability analysis and various bifurcations which are no more than codim two of the nonlinear feedback Block equations. The calculations are primarily in virtue of MATCONT, which is a software package based on MATLAB-GUI for the interactive numerical study of nonlinear and parametrized dyamical systems by continuation algorithm. It can compute curves of equilibria, limit points, Hopf points, limit cycles, period doubling bifurcation points of limit cycles, flip and torus bifurcation of limit cycles, periodic orbits and homoclinic orbits,etc. One or two parameters could be actived and the range of the parameters in bifurcation graphs is widely, so more complicated bifurcations could be found.Some basic conceptions of nonlinear dynamics such as existence, uniqueness, continuation of solution refer to differential equations and orbit, phase protrait, invariant set, equilibrium, limit circle, comdimension, manifold, structural stability, normal form refer to bifurcation theory are introduced in chapter two. Before bifurcations of nonlinear Block equations solving, the classifications and criterions of local bifurcations of nondegeneracy and transversality finite continuous systems whose codimensions are no more than two are introduced. Then the principle of numerical algorithms of nonlinear equations and main bifurcation softwares are introduced by comparison method in chapter three.At first, the nonlinear dimensionless feedback Block equations are derivated, then the analytical equilibria and stability regions of the equations are obtained. After stability analysis, integral of the equations are calculated by fixing value of parameters, the results as initial values will be for later continuation computing. Two parametersγandψare selected as free parameter respectivly, fold and Hopf bifurcations are got on codimension one. Then the Hopf bifurcation is selected as initial point for continuation of limit cycle on codimension one, fold bifurcation and flip bifurcation of limit cycle are found,γandψtogether as free parameters for continuation computing of equilibria and codimension two bifurcations such as Cusp, Fold-Hopf, Double-Zero bifurcation are found. It is obvious that totally seven bifurcations are found. Those results of nonlinear block equation have important and practical significance for understand and application of NMR(Nuclear Magnetic Resonance).