When difference equations are employed, the theory is called discrete dynamical systems.

## Deep learning for universal linear embeddings of nonlinear dynamics

When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a cantor set , one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations. This theory deals with the long-term qualitative behavior of dynamical systems, [1] and studies the nature of, and when possible the solutions of, the equations of motion of systems that are often primarily mechanical or otherwise physical in nature, such as planetary orbits and the behaviour of electronic circuits , as well as systems that arise in biology , economics , and elsewhere.

Much of modern research is focused on the study of chaotic systems. This field of study is also called just dynamical systems , mathematical dynamical systems theory or the mathematical theory of dynamical systems. Dynamical systems theory and chaos theory deal with the long-term qualitative behavior of dynamical systems.

Here, the focus is not on finding precise solutions to the equations defining the dynamical system which is often hopeless , but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible steady states? An important goal is to describe the fixed points, or steady states of a given dynamical system; these are values of the variable that don't change over time. Some of these fixed points are attractive , meaning that if the system starts out in a nearby state, it converges towards the fixed point.

Similarly, one is interested in periodic points , states of the system that repeat after several timesteps. Periodic points can also be attractive.

## Deep learning for universal linear embeddings of nonlinear dynamics

Sharkovskii's theorem is an interesting statement about the number of periodic points of a one-dimensional discrete dynamical system. Even simple nonlinear dynamical systems often exhibit seemingly random behavior that has been called chaos.

The concept of dynamical systems theory has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is given implicitly by a relation that gives the state of the system only a short time into the future. Before the advent of fast computing machines , solving a dynamical system required sophisticated mathematical techniques and could only be accomplished for a small class of dynamical systems.

The dynamical system concept is a mathematical formalization for any fixed "rule" that describes the time dependence of a point's position in its ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each spring in a lake. A dynamical system has a state determined by a collection of real numbers , or more generally by a set of points in an appropriate state space. Small changes in the state of the system correspond to small changes in the numbers.

The numbers are also the coordinates of a geometrical space—a manifold.

The evolution rule of the dynamical system is a fixed rule that describes what future states follow from the current state. The rule may be deterministic for a given time interval only one future state follows from the current state or stochastic the evolution of the state is subject to random shocks. Dynamicism , also termed the dynamic hypothesis or the dynamic hypothesis in cognitive science or dynamic cognition , is a new approach in cognitive science exemplified by the work of philosopher Tim van Gelder. It argues that differential equations are more suited to modelling cognition than more traditional computer models.

In mathematics , a nonlinear system is a system that is not linear —i. A nonhomogeneous system, which is linear apart from the presence of a function of the independent variables , is nonlinear according to a strict definition, but such systems are usually studied alongside linear systems, because they can be transformed to a linear system as long as a particular solution is known.

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Show all. This set of allowed solutions is referred to as the behavior of the system. Thus, the key is the behavior and not the representation.

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## Power converter circuits: A hybrid dynamical case

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