Application Of Clean Numerical Simulation In Chaotic Dynamic Systems

The physical world is complex and the relationships between objects are always nonlinear,such as turbulence,nonlinear vibration,waves and so on.Chaos is a typical phenomenon in nonlinear dynamic systems.Especially,chaos theory is considered as the third scientific evolution after relativity theory and quantum mechanics.Therefore,the study of chaotic dynamic systems has significant meanings in science and engineering.Because there is no analytic solutions for chaotic dynamic systems,numerical simulation is widely used to investigate chaotic dynamic systems.As is well known,there always are numerical noises(i.e.truncation error and round-off error)in the simulation of differential equations.The chaotic dynamic systems are sensitive to initial condition and numerical noise,therefore,the reliable computation of chaotic dynamic systems is a challenge problem.In this thesis,we gain reliable computation of chaotic dynamic systems by means of clean numerical simulation(CNS).The main work of the thesis is as follows.Firstly,we gain reliable numerical results for Hénon-Heiles system,three-body problem and the dynamic systems derived from Rayleigh-Bénard convection by means of clean numerical simulation(CNS)and compare these results with those obtained by traditional numerical methods with double precision.For Hénon-Heiles system and three-body problem,it is found that the traditional numerical methods in double precision cannot give not only reliable trajectories but also reliable Fourier power spectra and autocorrelation functions.In addition,for the dynamic systems derived from Rayleigh-Bénard convection,it is found that even statistic properties of chaotic systems can not be correctly obtained by means of traditional numerical algorithms in double precision,as long as these statistics are time dependent.Secondly,10000 samples of reliable(convergent)numerical simulations of a chaotic three-body system indicate that,without any external disturbance,the microscopic inherent uncertainty due to physical fluctuation of initial positions of the three-body system enlarges exponentially into macroscopic randomness.This indicates that the macroscopic randomness of the chaotic three-body system is self-excited,say,without any external force or disturbances,from the inherent micro-level uncertainty.It implies that chaos might be a bridge between the micro-level uncertainty and macroscopic randomness.Lastly,we find more than 2000 new periodic orbits for the three-body problem.In the 300 years since the three-body problem was first recognized,only three families of periodic solutions had been found,until 2013 when [Phys.Rev.Lett.110,114301(2013)] made a breakthrough to numerically find 13 new distinct periodic orbits.In this thesis,we find more than 600 Newtonian periodic planar collisionless orbits of threebody system with equal mass,over 1200 periodic orbits of three-body problem with unequal mass and more than 300 collisionless periodic orbits of the free-fall threebody problem by means of grid search method,Newton-Raphson method and clean numerical simulation(CNS).It is found that there should exist the general Kepler's third law for periodic three-body problem,i.e.,the scale-invariant average period is close to a constant.It has been traditionally believed that triple systems are often unstable if they are non-hierarchical.However,all of our new periodic orbits are in non-hierarchical configurations,but 28 periodic orbits are linearly stable.This might inspire the longterm astronomical observation of stable non-hierarchical triple systems in practice.