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Finite-dimensional linear systems are never chaotic; for a dynamical system to display chaotic behavior, it must be either nonlinear or infinite-dimensional. The Poincaré–Bendixson theorem states that a two-dimensional differential equation has very regular behavior.
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. [1] [2] Nonlinear problems are of interest to engineers, biologists, [3] [4] [5] physicists, [6] [7] mathematicians, and many other scientists since most systems are inherently nonlinear in nature. [8]
Much of modern research is focused on the study of chaotic systems and bizarre systems. This field of study is also called just dynamical systems, mathematical dynamical systems theory or the mathematical theory of dynamical systems. A chaotic solution of the Lorenz system, which is an example of a non-linear dynamical system.
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.
The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz
Chaos is not peculiar to non-linear systems alone and it can also be exhibited by infinite dimensional linear systems. [11] As mentioned above, the logistic map itself is an ordinary quadratic function. An important question in terms of dynamical systems is how the behavior of the trajectory changes when the parameter r changes.
Chaotic maps often occur in the study of dynamical systems. Chaotic maps and iterated functions often generate fractals . Some fractals are studied as objects themselves, as sets rather than in terms of the maps that generate them.
Period doubling has been observed in a number of experimental systems. [7] There is also experimental evidence of period-doubling cascades. For example, sequences of 4 period doublings have been observed in the dynamics of convection rolls in water and mercury. [8] [9] Similarly, 4-5 doublings have been observed in certain nonlinear electronic ...