Polynomial Approach and Non-linear Analysis for a Traffic Fundamental Diagram

Vehicular traffic can be modelled as a dynamic discrete form. As in many dynamic systems, the parameters modelling traffic can produce a number of different trajectories or orbits, and it is possible to depict different flow situations, including chaotic ones. In this paper, an approach to the wellknown density-flow fundamental diagram is suggested, using an analytical polynomial technique, in which coefficients are taken from significant values acting as the parameters of the traffic model. Depending on the values of these parameters, it can be seen how the traffic flow changes from stable endpoints to chaotic trajectories, with proper analysis in their stability features

A chemical representation of a chaotic system with a unique stable equilibrium point

"In this paper we present a chemical representation of a chaotic system with only one stable equilibrium point. The approach invokes cooperative catalysis and slow-fast reactions, primarily. The obtained chemical based chaotic dynamical system preserves the eigenvalues of the unique and stable equilibrium point along with the Lyapunov’s dimension and exponents of the original one.

Preservation of Synchronization Using a Tracy‐Singh Product in the Transformation on Their Linear Matrix

Preservation is related to local asymptotic stability in nonlinear systems by using dynamical systems tools. It is known that a system, which is stable, asymptotically stable, or unstable at origin, through a transformation can remain stable, asymptotically stable, or unstable. Some systems permit partition of its nonlinear equation in a linear and nonlinear part. Some authors have stated that such systems preserve their local asymptotic stability through the transformations on their linear part. The preservation of synchronization is a typical application of these types of tools and it is considered an interesting topic by scientific community. This chapter is devoted to extend the methodology of the dynamical systems through a partition in the linear part and the nonlinear part, transforming the linear part using the Tracy-Singh product in the Jacobian matrix. This methodology preserves the structure of signs through the real part of eigenvalues of the Jacobian matrix of the dynamical systems in their equilibrium points. The principal part of this methodology is that it permits to extend the fundamental theorems of the dynamical systems, given a linear transformation. The results allow us to infer the hyperbolicity, the stability and the synchronization of transformed systems of higher dimension

Polynomial Approach and Non-linear Analysis for a Traffic Fundamental Diagram

Vehicular traffic can be modelled as a dynamic discrete form. As in many dynamic systems, the parameters modelling traffic can produce a number of different trajectories or orbits, and it is possible to depict different flow situations, including chaotic ones. In this paper, an approach to the wellknown density-flow fundamental diagram is suggested, using an analytical polynomial technique, in which coefficients are taken from significant values acting as the parameters of the traffic model. Depending on the values of these parameters, it can be seen how the traffic flow changes from stable endpoints to chaotic trajectories, with proper analysis in their stability features