Generalised energy conservation law for chaotic phenomena

Abstract

Chaotic phenomena are increasingly being observed in all fields of nature, where investigations reveal that a natural phenomenon exhibits nonlinearities and attempts to reveal their deep underlying mechanisms. Chaos is normally understood as "a state of disorder", for which there is as yet no universally accepted mathematical definition. A commonly used concept states that, for a dynamical system to be classified as chaotic, it must have the following properties: be sensitive to initial conditions, show topological transitivity, have densely periodical orbits etc. Revealing the rules that govern chaotic motion is thus an important unsolved task for exploring nature. We present herein a generalised energy conservation law governing chaotic phenomena. Based on two scalar variables, viz. generalised potential and kinetic energies defined in the phase space describing nonlinear dynamical systems, we find that chaotic motion is periodic motion with infinite time period whose time-averaged generalised potential and kinetic energies are conserved over its time period. This implies that, as the averaging time is increased, the time-averaged generalised potential and kinetic energies tend to constants while the time-averaged energy flows, i.e., their rates of change with time, tend to zero. Numerical simulations on reported chaotic motions, such as the forced van der Pol system, forced Duffing system, forced smooth and discontinuous oscillator, Lorenz's system, and Rössler's system, show the above conclusions to be correct according to the results presented herein. This discovery may indicate that chaotic phenomena in nature could be controlled because, even though their instantaneous states are disordered, their long-time averages can be predicted.

Energy flow characteristics of friction-induced nonlinear vibrations in a water-lubricated bearing-shaft coupled system

“…Xing [28,29] has discovered that a chaotic motion of nonlinear dynamical system can be considered as a periodical motion with an infinte long time period and the following integrations hold (34c)…”

“…To reveal nonlinear behavior of nonlinear dynamical systems, such as as discussed in the books by Guckenheimer and Holmes [24], Thompson and Stewart [25], Chen et al [26] and Liu and Chen [27], Xing [28] developed a genralised energy flow theory to investigate nonlinear dynamical systems governed by ordinary differential equation in phase space. Important nonlinear phenomena such as, stabilities, periodical orbits, bifurcations and chaos can be revealed from the energy flow behavors of the systems [29]. The aim and motivation of this paper are to investigate the 2-DOF nonlinear model of the stern-tube bearing proposed in Ref.…”

Theoretical investigations on lattice Boltzmann method: an amended MBD and improved LBM

“…The paper is a theoretical document written exactly based on the famous Boltzmann H-theorem [6] of the statistical mechanics, the laws of continuum mechanics [113][114][115], the energy flow theory [116][117][118][119][120], and the variational method in mathematical analysis. Section 1 from reading historical publications reveals the existing problems of LBM theory, that is the total internal energy is not involved in current EDF causing some theoretical and numerical problems.…”