Solving Complex System Dynamical Averages Using Periodic Orbit Theory

In complex dynamical systems,the vast majority of systems are chaotic and measurable leading to the fact that it is impossible to solve their kinetic parameters accurately.We usually calculate the average of some meaningful physical quantities to reflect the general properties of system.However,traditional computational methods are not applicable in complex nonlinear systems.So we introduce the periodic orbit theory to solve the average value of complex systems.In periodic orbit theory,for uniform systems with hyperbolicity,the circle expansion of periodic orbit theory is rapidly converged,which means the long period orbits are followed with several basic cycle structures composed of short period orbits,so the motion of the dynamical system can be described by some short period orbits,the average of the system physical can be calculated by intercepting a small number of short unstable periodic orbits.However,the convergence of circle expansions in systems with nonhyperbolic will be poor,which result is exist the critical point or intermittency phenomenon in partial dynamical systems.For dynamical systems with critical points,we eliminate the effect of the critical point by conjugate map and improve the convergence of the circle expand.For dynamical systems with intermittency,the critical periodic orbit in phase space leads to slow converge speed,and we accelerate the convergence of the circle expansion by constructing a new map with the same natural measure as the original map without losing any dynamical information.