Finding the salient patterns in chaotic data has been the holy grail of Chaos Theory. Examples of chaotic data include the fluctuations of the stock market, weather, and many other natural systems. Real world data has proven to be extremely difficult to predict due to it high dimensionality and the potential for noise. It has been shown that artificial neural networks have been able to accurate...

The dynamics of the growth of linear disturbances to a chaotic basic state is analyzed in an asymptotic model of weakly nonlinear, baroclinic wave-mean interaction. In this model, an ordinary differential equation for the wave amplitude is coupled to a partial differential equation for the zonal flow correction. The leading Lyapunov vector is nearly parallel to the leading Floquet vector φ1 of ...

In [3] Conley showed that the state-space of a dynamical system can be decomposed into a gradient-like part and a chain-recurrent part, and that this decomposition is characterized by a so-called complete Lyapunov function for the system. In [14] Kalies, Mischaikow, and VanderVorst proposed a combinatorial method to compute discrete approximations to such complete Lyapunov functions. Their appr...

This paper presents procedural guidelines for the construction of discontinuous state feedback controllers for driftless, kinematic nonholonomic systems, with extensions to a class of dynamic nonholonomic systems with drift. Given an ndimensional kinematic nonholonomic system subject to κ Pfaffian constraints, system states are partitioned into “leafwise” and “transverse,” based on the structur...

In this paper we construct a trajectory tracking controller for a quadrotor system by finding a coordinate change which transforms the quadrotor’s vector field into that of a thust propelled system. In a thrust propelled system, the goal is to stabilize its position around the origin, while the system is actuated by a one dimensional acceleration/thrust along a direction vector, by a time-varyi...

This paper investigates set-valued differential equations with fractional-like Hukuhara derivatives. Firstly, a novel comparison principle is given by introducing the upper quasi-monotone increasing functions. Then, stability criteria of Lipschitz and practical such different initial time are obtained via new vector Lyapunov

We consider stochastic differential equations in d-dimensional Euclidean space driven by an m-dimensional Wiener process, determined by the drift vector field f0 and the diffusion vector fields f1, · · · , fm, and investigate the existence of global random attractors for the associated flows φ. For this purpose φ is decomposed into a stationary diffeomorphism Φ given by the stochastic different...

We consider the multi-task coordination problem for multi-agent systems under the following objectives: 1. collision avoidance; 2. connectivity maintenance; 3. convergence to desired destinations. The paper focuses on the safety guaranteed region of multi-task coordination (SG-RMTC), i.e., the set of initial states from which all trajectories converge to the desired configuration, while at the ...

Introduction. The primary purpose of this note is to construct a C* Lyapunov function for a continuous vector field which has an asymptotically stable invariant set. Since there already exist techniques for constructing continuous Lyapunov functions with continuous derivatives by the vector field, our problem is simply to smooth such a function in such a way that the desired derivative properti...