Chaotic Dynamics in Multidimensional Transition States

The crossing of a transition state in a multidimensional reactive system is mediated by invariant geometric objects in phase space: An invariant hyper-sphere that represents the transition state itself and invariant hyper-cylinders that channel the system towards and away from the transition state. The existence of these structures can only be guaranteed if the invariant hyper-sphere is normally hyperbolic, i.e., the dynamics within the transition state is not too strongly chaotic. We study the dynamics within the transition state for the hydrogen exchange reaction in three degrees of freedom. As the energy increases, the dynamics within the transition state becomes increasingly chaotic. We find that the transition state first looses and then, surprisingly, regains its normal hyperbolicity. The important phase space structures of transition state theory will therefore exist at most energies above the threshold

Numerical investigation of chaotic dynamics in multidimensional transition states

Many chemical reactions can be described as the crossing of an energetic barrier. This process is mediated by an invariant object in phase space. One can construct a normally hyperbolic invariant manifold (NHIM) of the reactive dynamical system which is an invariant sphere that can be considered as the geometric representation of the transition state itself. The NHIM has invariant cylinders (reaction channels) attached to it. This invariant geometric structure survives as long as the invariant sphere is normally hyperbolic. We applied this theory to the hydrogen exchange reaction in three degrees of freedom in order to figure out the reason of the transition state theory (TST) failure. Energies high above the reaction threshold, the dynamics within the transition state becomes partially chaotic. We have found that the invariant sphere first ceases to be normally hyperbolic at fairly low energies. Surprisingly normal hyperbolicity is then restored and the invariant sphere remains normally hyperbolic even at very high energies. This observation shows two different energy values for the breakdown of the TST and the breakdown of the NHIM. This leads to seek another phase space object that is related to the breakdown of the TST. Using theory of the dividing surface including reactive islands (RIs), we can investigate such an object. We found out that the first nonreactive trajectory has been found at the same energy values for both collinear and full systems, and coincides with the first bifurcation of periodic orbit dividing surface (PODS) at the collinear configuration. The bifurcation creates the unstable periodic orbit (UPO). Indeed, the new PODS (UPO) is the reason for the TST failure. The manifolds (stable and centre-stable) of the UPO clarify these expectations by intersecting the dividing surface at the boundary of the reactive island (on the collinear and the three (full) systems, respectively)

Chaotic dynamics in multidimensional transition states

The crossing of a transition state in a multidimensional reactive system is mediated by invariantgeometric objects in phase space: An invariant hyper-sphere that represents the transition stateitself and invariant hyper-cylinders that channel the system towards and away from the transitionstate. The existence of these structures can only be guaranteed if the invariant hyper-sphere isnormally hyperbolic, i.e., the dynamics within the transition state is not too strongly chaotic. We study thedynamics within thetransition state for the hydrogen exchangereaction in three degrees of freedom. As the energy increases, the dynamics within the transition statebecomes increasingly chaotic. We find that the transition state first looses and then, surprisingly,regains its normal hyperbolicity. The important phase space structures of transition state theory will, therefore,exist at most energies above the threshold

Stability analysis of chaotic generalized Lotka-Volterra system via active compound difference anti-synchronization method

This work deals with a systematic approach for the investigation of compound difference anti-synchronization (CDAS) scheme among chaotic generalized Lotka-Volterra biological systems (GLVBSs). First, an active control strategy (ACS) of nonlinear type is described which is specifically based on Lyapunov's stability analysis (LSA) and master-slave framework. In addition, the biological control law having nonlinear expression is constructed for attaining asymptotic stability pattern for the error dynamics of the discussed GLVBSs. Also, simulation results through MATLAB environment are executed for illustrating the efficacy and correctness of considered CDAS approach. Remarkably, our attained analytical outcomes have been in outstanding conformity with the numerical outcomes. The investigated CDAS strategy has numerous significant applications to the fields of encryption and secure communication

Synchronized Chaos of a Three-Dimensional System with Quadratic Terms

In this paper, a novel chaotic new three-dimensional system has been studied by Zhang et al. in 2012. In the system, there are three control parameters and three different nonlinear terms which governed equations. Zhang et al. studied elementary (preliminary) dynamic properties of the chaotic new three-dimensional system by means of bifurcation diagram, maximum Lyapunov exponent, phase portraits, dynamics behaviors by changing some parameters etc., using all possible theoretical analysis and numerical simulation. In this paper, we have demonstrated its complete synchronization. The proposed results are verified by numerical simulations

Analytical Solution to Normal Forms of Hamiltonian Systems

The idea of the normalisation of the Hamiltonian system is to simplify the system by transforming Hamiltonian canonically to an easy system. It is under symplectic conditions that the Hamiltonian is preserved under a specific transformation—the so-called Lie transformation. In this review, we will show how to compute the normal form for the Hamiltonian, including computing the general function analytically. A clear example has been studied to illustrate the normal form theory, which can be used as a guide for arbitrary problems

Existence of Positive Solutions of Nonlocal <i>p</i>(<i>x</i>)-Kirchhoff Evolutionary Systems via Sub-Super Solutions Concept

Motivated by the idea which has been introduced by Boulaaras and Guefaifia (Math. Meth. Appl. Sci. 41 (2018), 5203&#8315;5210 and, by Afrouzi and Shakeri (Afr. Mat. (2015) 26:159&#8315;168) combined with some properties of Kirchhoff type operators, we prove the existence of positive solutions for a class of nonlocal p x -Kirchhoff evolutionary systems by using the sub and super solutions concept

A New Mathematical Model and its Application in the Pollution of Air and Water: An Application of Virtual Experience in Qassim Province in Kingdom of Saudi Arabia

In this paper, we present a numerical model of one-dimensional equations of transport equation, where we apply the model in a hypothetical example and compare the results of our model with a virtual experience which deals with the concentration of certain pollutants and their speed diffusion in water in Qassim province in Kingdom of Saudi Arabia

Existence of Positive Solutions for a Class of px,qx-Laplacian Elliptic Systems with Multiplication of Two Separate Functions

The paper deals with the study of the existence of weak positive solutions for a new class of the system of elliptic differential equations with respect to the symmetry conditions and the right hand side which has been defined as multiplication of two separate functions by using the sub-supersolutions method (1991 Mathematics Subject Classification: 35J60, 35B30, and 35B40)