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Depending on the value of μ, the tent map demonstrates a range of dynamical behaviour ranging from predictable to chaotic. If μ is less than 1 the point x = 0 is an attractive fixed point of the system for all initial values of x i.e. the system will converge towards x = 0 from any initial value of x.
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?", or "Does ...
The definition for discrete-time systems is almost identical to that for continuous-time systems. The definition below provides this, using an alternate language commonly used in more mathematical texts. Let (X, d) be a metric space and f : X → X a continuous function. A point x in X is said to be Lyapunov stable, if,
In mathematics, the Hénon map, sometimes called Hénon–Pomeau attractor/map, [1] is a discrete-time dynamical system. It is one of the most studied examples of dynamical systems that exhibit chaotic behavior. The Hénon map takes a point (x n, y n) in the plane and maps it to a new point
A discrete dynamical system, discrete-time dynamical system is a tuple (T, M, Φ), where M is a manifold locally diffeomorphic to a Banach space, and Φ is a function. When T is taken to be the integers, it is a cascade or a map. If T is restricted to the non-negative integers we call the system a semi-cascade. [14]
The state vector (vector of state variables) representing the current state of a discrete-time system (i.e. digital system) is [], where n is the discrete point in time at which the system is being evaluated. The discrete-time state equations are [+] = [] + [], which describes the next state of the system (x[n+1]) with respect to current state ...
In the case of a marble on top of an inverted bowl (a hill), that point at the top of the bowl (hill) is a fixed point (equilibrium), but not an attractor (unstable equilibrium). In addition, physical dynamic systems with at least one fixed point invariably have multiple fixed points and attractors due to the reality of dynamics in the physical ...
A "linear time-invariant differential system" is a dynamical system = (,,) whose behavior is the solution set of a system of constant coefficient linear ordinary differential equations (/) =, where is a matrix of polynomials with real coefficients.