Generalised energy conservation law of chaotic phenomena

Chaotic phenomena are more and more observed in any fields of nature, when investigations on natural phenomena enter into nonlinearities to reveal their deep mechanisms. Chaos is normally understood as “a state of disorder”, for which until now there is no universally accepted a mathematical definition. Commonly used concept says that, for a dynamical system to be classified as chaotic, it must have the following properties: sensitive to initial conditions; topological transitivity; densely periodical orbits etc. To reveal the rules governing chaotic motions is an important unsolved task exploring nature. Here we show a generalised energy conservation law governing chaotic phenomena. Based on the two scalar variables: generalised potential and kinetic energies defined in the phase space describing nonlinear dynamical systems, we found that a chaotic motion is a periodical motion with infinite time period and its time-averaged generalised potential and kinetic energies are in conservation over its time period. This implies that with the average time increasing, the time averaged generalised potential and kinetic energies tend constants while their time-averaged energy flows, their time change rates, tend zero. The numerical simulations on the reported chaotic motions: forced Van der Pol’s system, forced duffing’s system, forced SD oscillator, Lorenz’s system and Rössler’s system, have demonstrated the above conclusion is correct, of which the obtained curves are given in the paper. The discover may advice that any chaotic phenomena of nature could be controlled, although their instant states are in disorder but the long-time averages are predicated