# Nonlinear Orthonormal Bases for Dynamical Systems in 3D

• arvind-ipr
In summary, the conversation discusses the proposal of 27 scalar functions, 9 delay-coupled 3D unit vectors, 7 ortho-normal bases, 4 tensors, and a tensor E of varying orders, all of which are defined in terms of internal variables and a dimensionless real variable. These functions can be used to model complex dynamical systems, nonlinear dynamics, chaos, plasma waves, and turbulence. Additionally, the tensors can represent rotation, shear, and wave propagation.
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.

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.

## 1. What are nonlinear orthonormal bases for dynamical systems in 3D?

Nonlinear orthonormal bases are sets of three vectors that are both orthogonal (perpendicular) to each other and unit in length. These bases are used to represent dynamical systems in three-dimensional space, where the basis vectors can be rotated and scaled to accurately describe the behavior of the system over time.

## 2. How are nonlinear orthonormal bases used in 3D dynamical systems?

Nonlinear orthonormal bases are used in 3D dynamical systems to describe the behavior of a system at different points in time. By rotating and scaling the basis vectors, the system can be accurately represented as it evolves over time, providing insight into its behavior and dynamics.

## 3. What types of systems can be described using nonlinear orthonormal bases in 3D?

Nonlinear orthonormal bases are commonly used to describe complex systems that exhibit chaotic or nonlinear behavior, such as weather patterns, fluid flow, and biological systems. They can also be applied to mechanical systems and control systems to analyze their dynamics.

## 4. How are nonlinear orthonormal bases different from traditional bases?

Traditional bases, such as Cartesian coordinates, are linear and do not account for changes in the system over time. Nonlinear orthonormal bases, on the other hand, can adapt to the dynamics of the system and accurately represent its behavior at different points in time. This makes them more suitable for describing complex and nonlinear systems.

## 5. What are the benefits of using nonlinear orthonormal bases for dynamical systems in 3D?

Nonlinear orthonormal bases provide a powerful tool for analyzing and understanding complex systems. They can reveal patterns and behaviors that may not be apparent using traditional methods, and can also help identify key factors that influence the system's dynamics. Additionally, they can help with making predictions and controlling the behavior of the system.

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