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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]
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 ...
discrete: complex: 1: 0: acts on the Julia set for the squaring map. Complex cubic map: discrete: complex: 1: 2: Clifford fractal map [13] discrete: real: 2: 4: Degenerate Double Rotor map: De Jong fractal map [14] discrete: real: 2: 4: Delayed-Logistic system [15] discrete: real: 2: 1: Discretized circular Van der Pol system [16] discrete ...
Consider an autonomous nonlinear dynamical system ˙ = (()), =, where () denotes the system state vector, an open set containing the origin, and : is a continuous vector field on .
Together, the state and costate equations describe the Hamiltonian dynamical system (again analogous to but distinct from the Hamiltonian system in physics), the solution of which involves a two-point boundary value problem, given that there are boundary conditions involving two different points in time, the initial time (the differential ...
The logistic map is a discrete dynamical system defined by the quadratic difference equation: ... (This problem does not appear in the older Ricker model, ...
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.
Optimal control problem benchmark (Luus) with an integral objective, inequality, and differential constraint. Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. [1]