Search results
Results from the WOW.Com Content Network
For r between 1 and 3, the value 0 is still periodic but is not attracting, while the value is an attracting periodic point of period 1. With r greater than 3 but less than 1 + 6 , {\displaystyle 1+{\sqrt {6}},} there are a pair of period-2 points which together form an attracting sequence, as well as the non-attracting period-1 points ...
Sharkovskii's theorem states that if has a periodic point of least period , and precedes in the above ordering, then has also a periodic point of least period . One consequence is that if f {\displaystyle f} has only finitely many periodic points, then they must all have periods that are powers of two.
A (purely) periodic sequence (with period p), or a p-periodic sequence, is a sequence a 1, a 2, a 3, ... satisfying . a n+p = a n. for all values of n. [1] [2] [3] If a sequence is regarded as a function whose domain is the set of natural numbers, then a periodic sequence is simply a special type of periodic function.
This article describes periodic points of some complex quadratic maps.A map is a formula for computing a value of a variable based on its own previous value or values; a quadratic map is one that involves the previous value raised to the powers one and two; and a complex map is one in which the variable and the parameters are complex numbers.
In dynamical systems one would like to understand the distribution of fixed points or periodic points. A periodic point of a Hamiltonian diffeomorphism (of periodic ) is a point such that () =. A feature of Hamiltonian dynamics is that Hamiltonian diffeomorphisms tend to have infinitely many periodic points.
Stable periodic point: In this case, the Lyapunov exponent is negative. Aperiodic orbits: In this case, the Lyapunov exponent is positive. The region of stable periodic points that exists for r < is called a periodic window, or simply a window. If one looks at a chaotic region in an orbital diagram, the region of nonperiodic orbits looks like a ...
It is known in this case that F c (x) cannot have periodic points of period four, [4] five, [5] or six, [6] although the result for period six is contingent on the validity of the conjecture of Birch and Swinnerton-Dyer. Bjorn Poonen has conjectured that F c (x) cannot have rational periodic points of any period strictly larger than three. [7]
A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic (that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that is if there is only a finite number of nonzero digits), the decimal is said to be terminating, and is not considered as repeating.