Chaos Indused By The Diffusion Euquation In A Closed Loop

Infinite dimensional dynamical systems which are based on finite demensional dy-namical systems, have been researched for more than fifty years. The most importantresult of recent researches is discovering that the solutions of quite many PDEs with dissi-pative e?ect are consistent to the finite dimensional systems when the time is large. Forexample, global atteactors, inertial manifolds, attractor dimensions, Galerkin methodand their application are very important in the research.At present,th basic theorems ofinfinite dimensional dynamical systems have been established. As the development ofcomputational techniques and numerical methods, the applications of numerical simu-lation have help many researchers fin a lot of new results in bifurcation, chaos, solitonand fractal. This promoted the application of infinite dynamical systems in nonlin-ear sciences. As this kind of problems is very common in practical applications, manyresearchers are attracted to research these systems.In this paper, we investigate the ?owing law of the liquid inside a closed loop, thatis the distribution of temperature and velocity of the liquid. With the help of theoreticalanalysis and numerical simulation, we can show that the ?uid in a closed loop exhibitsthe phenomenon of chaos.In the first chapter, we introduce some definitions and theorems about the infinitedynamical systems, including global attractors, analytical semigroup, inertial manifolds.And in the last part of chaper 1, we give the system model discussed in this paper. In thesecond chapter, we give the well-posedness and boundedness of solutions to the systemin two cases, that is with di?usion and with no di?usion. With the help of ninertialmanifold technique, we prove the existence of the global attractor. And we give thedetail analysis to the asymptotical behavior of solutions and whenεis small enough, thebehavior of solution by Fourier expantion. In the third chapter, we give the result of thenumerical simulation,and can find the phenomenon of chaos.