Nonlinear Orthonormal Bases for Dynamical Systems in 3D

[arvind-ipr]

I propose 27 scalar functions {fn : n = 1,2,…,27}, 9 delay-coupled 3D unit vectors {eij : i,j = 1,2,3} of general periodic nature, 7 ortho-normal bases {Zk : k = 0,1,2,…,6} each having three 3D unit vectors, 4 tensors {Y3,Y4,Y5,Y6} of order 2 and a tensor E of order 3. Z3-Z6 and E are defined in terms of linearly independent nonlinear vector functions eij of ψ. The rectangular components of eij and elements of orthogonal matrices Y3-Y6 are defined in terms of fn. All fn are well defined explicit functions of internal variables θ, φ, η1, η2 which themselves are simple/composite functions of a dimensionless non-negative real variable ψ where ψ may be either independent or linear function of independent time variable t. The constant bases Z0, Z1 and linear basis Z2 are special cases of Z3 which have 3 key nonlinear unit vectors ei1. 27 fn include 2 well-known sinusoidal, 21 new linearly independent bounded non-sinusoidal periodic/non-periodic oscillatory functions and 4 trivial Zero polynomials. The periodic nature (and common fundamental period in periodic case) of all 21 non-sinusoidal functions and 9 eij depend on a free periodic/non-periodic sequence s = {Am: Am ε [0, 1], m ε W}.
The nonlinear orthonormal bases Z3-Z6 may be used in modeling a wide range of deterministic complex dynamical systems involving nonlinear dynamics, chaos, nonlinear plasma waves and turbulence. In particular, one can model collective nonlinear dynamics of 3 delay-coupled 3D anharmonic oscillators in any bounded region of R3 and deterministic Brownian motion of an arbitrary large number of particles in fractal geometry confined in any bounded region of R3.

 

[Valerie Park]

The tensors Y3-Y6 may be used to represent arbitrary 3D rotation, shear and twist operations. The tensor E may be used in modeling nonlinear wave propagation, diffraction, interference, refraction and reflection of light.