The Lorenz system is one of the most widely studied of the many chaotic systems now known, and it is the prototypical example of sensitive dependence on initial conditions (the butterfly effect). For the standard parameters used by Lorenz, there is a single symmetric double-wing chaotic attractor that resembles a butterfly. In this paper, the dynamical behavior of the Lorenz system is examined ...

A new technique for obtaining rigorous results concerning the global dynamics of nonlinear systems is described. The technique combines abstract existence results based on the Conley index theory with computer-assisted computations. As an application of these methods it is proven that for an explicit parameter value the Lorenz equations exhibit chaotic dynamics. Introduction The purpose of this...

Title: The fundamental solution of compressible and incompressible fluid flow past a rotating obstacle Abstract: We consider the flow of either an incompressible or a compressible fluid around or past a rotating rigid body in the whole space R 3. Using a global coordinate transform and a linearization the problem reduces to a linear PDE system in a time-independent domain which may admit statio...

There are two parts in this dissertation. The backward stochastic Lorenz system is studied in the first part. Suitable a priori estimates for adapted solutions of the backward stochastic Lorenz system are obtained. The existence and uniqueness of solutions is shown by the use of suitable truncations and approximations. The continuity of the adapted solutions with respect to the terminal data is...

In this paper, at first we derive a family of maximum Tsallis entropy distributions under optional side conditions on the mean income and the Gini index. Furthermore, corresponding with these distributions a family of Lorenz curves compatible with the optional side conditions is generated. Meanwhile, we show that our results reduce to Shannon entropy as $beta$ tends to one. Finally, by using ac...

We formulate a spherical harmonically decomposed 1þ 1 scheme to self-consistently evolve the trajectory of a point particle and its gravitational metric perturbation to a Schwarzschild background spacetime. Following the work of Moncrief, we write down an action for perturbations in spacetime geometry, combine that with the action for a point particle moving through this spacetime, and then obt...

Consider a continuous dynamical system for which partial information about its current state is observed at a sequence of discrete times. Discrete data assimilation inserts these observational measurements of the reference dynamical system into an approximate solution by means of an impulsive forcing. In this way the approximating solution is coupled to the reference solution at a discrete sequ...

In this paper, we study chaotic synchronization in lattices of coupled Lorenz equations with Neumann or periodic boundary condition. Three different coupling configurations in the single xi-, yior zi-component are considered. Synchronization is affected by coupling rules. We prove that synchronization occurs for either xior yi-component coupling provided the coupling coefficient is sufficiently...