The book employs oscillatory dynamical systems to represent the Universe mathematically via constructing classical and quantum theory of damped oscillators. It further discusses isotropic and homogeneous metrics in the Friedman-Robertson-Walker Universe and shows their equivalence to non-stationary oscillators. The wide class of exactly solvable damped oscillator models with variable parameters is associated with classical special functions of mathematical physics. Combining principles with observations in an easy to follow way, it inspires further thinking for mathematicians and physicists. Contents Part I: Dissipative geometry and general relativity theory Pseudo-Riemannian geometry and general relativity Dynamics of universe models Anisotropic and homogeneous universe models Metric waves in a nonstationary universe and dissipative oscillator Bosonic and fermionic models of a Friedman-Robertson-Walker universe Time dependent constants in an oscillatory universe Part II: Variational principle for time dependent oscillations and dissipations Lagrangian and Hamilton descriptions Damped oscillator: classical and quantum theory Sturm-Liouville problem as a damped oscillator with time dependent damping and frequency Riccati representation of time dependent damped oscillators Quantization of the harmonic oscillator with time dependent parameters

Nonlinear Evolution Equations and Dynamical Systems (NEEDS) provides a presentation of the state of the art. But for some exceptions, the contributions are intentionally brief to give only the gist of the methods, proofs, etc. including references to the relevant literature. This gives a handy overview of current research activities. Hence, the book should be equally useful to the senior researcher as well as the colleague just entering the field. Topics treated: One- and multidimensional (integrable) models, geometric and algebraic methods, quantum field theory, applications to nonlinear optics, condensed matter physics, oceanography, and many others. Further keywords: Hirota bilinearity, Hamiltonians, Toda lattice, multi-dimensional inverse scattering, bifurcations, dromions, polynomial solutions, Ermakov systems, computer algebra, symplectic operators, (quantum) superalgebras, groups, Ising model.