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Dynamical neuroscience describes the non-linear dynamics at many levels of the brain from single neural cells [3] to cognitive processes, sleep states and the behavior of neurons in large-scale neuronal simulation. [4] Neurons have been modeled as nonlinear systems for decades, but dynamical systems are not constrained to neurons.
However, real-world systems are often nonlinear and multidimensional, in some instances rendering explicit equation-based modeling problematic. Empirical models, which infer patterns and associations from the data instead of using hypothesized equations, represent a natural and flexible framework for modeling complex dynamics.
Sparse identification of nonlinear dynamics (SINDy) is a data-driven algorithm for obtaining dynamical systems from data. [1] Given a series of snapshots of a dynamical system and its corresponding time derivatives, SINDy performs a sparsity-promoting regression (such as LASSO) on a library of nonlinear candidate functions of the snapshots against the derivatives to find the governing equations.
The Duffing equation can be seen as describing the oscillations of a mass attached to a nonlinear spring and a linear damper. The restoring force provided by the nonlinear spring is then α x + β x 3 . {\displaystyle \alpha x+\beta x^{3}.}
A coupled map lattice (CML) is a dynamical system that models the behavior of nonlinear systems (especially partial differential equations).They are predominantly used to qualitatively study the chaotic dynamics of spatially extended systems.
While the method converges under general conditions, it typically makes slower progress than competing methods. Nonetheless, the study of relaxation methods remains a core part of linear algebra, because the transformations of relaxation theory provide excellent preconditioners for new methods.
System identification is a method of identifying or measuring the mathematical model of a system from measurements of the system inputs and outputs. The applications of system identification include any system where the inputs and outputs can be measured and include industrial processes, control systems, economic data, biology and the life sciences, medicine, social systems and many more.
MINPACK is a library of FORTRAN subroutines for the solving of systems of nonlinear equations, or the least squares minimization of the residual of a set of linear or nonlinear equations. The NAG Fortran Library is a collection of mathematical and statistical routines for Fortran. NOVAS is a software library for astrometry-related numerical ...