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Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems.
A real dynamical system, real-time dynamical system, continuous time dynamical system, or flow is a tuple (T, M, Φ) with T an open interval in the real numbers R, M a manifold locally diffeomorphic to a Banach space, and Φ a continuous function. If Φ is continuously differentiable we say the system is a differentiable dynamical system.
A continuous dynamical system is a Lie point symmetry of if, and only if, sends every orbit of to an orbit. Hence, the infinitesimal generator δ S {\displaystyle \delta _{\mathcal {S}}} satisfies the following relation [ 8 ] based on Lie bracket :
The Rössler attractor (/ ˈ r ɒ s l ər /) is the attractor for the Rössler system, a system of three non-linear ordinary differential equations originally studied by Otto Rössler in the 1970s. [1] [2] These differential equations define a continuous-time dynamical system that exhibits chaotic dynamics associated with the fractal properties ...
In the mathematical area of bifurcation theory a saddle-node bifurcation, tangential bifurcation or fold bifurcation is a local bifurcation in which two fixed points (or equilibria) of a dynamical system collide and annihilate each other. The term 'saddle-node bifurcation' is most often used in reference to continuous dynamical systems.
On a torus, for example, it is possible to have a recurrent non-periodic orbit, [3] and three-dimensional systems may have strange attractors. Nevertheless, it is possible to classify the minimal sets of continuous dynamical systems on any two-dimensional compact and connected manifold due to a generalization of Arthur J. Schwartz. [4] [5]
A (real-world) dynamic system may be continuous or discrete. Continuous dynamic systems (like physical systems with material objects moving in space) are characterized by state variables the values of which change continuously, while the state variable values of discrete dynamic systems (like predator-prey ecosystems) "jump", that is, they are ...
A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior – a system that can both flow (described by a differential equation) and jump (described by a state machine, automaton, or a difference equation). [1]