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 ...
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.
Over 25 years after the publication of his first paper, Jungck defined additional conditions under which and will have a common fixed point, based on the notions of periodic points and the coincidence set of the functions, that is, the values for which () = (). [27]
A periodic point z of f is called a saddle periodic point if, for a positive integer r such that () =, at least one eigenvalue of the derivative of on the tangent space at z has absolute value less than 1, at least one has absolute value greater than 1, and none has absolute value equal to 1.
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.
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.
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 ...
In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation. Specifically, for functions, a fixed point is an element that is mapped to itself by the function. Any set of fixed points of a transformation is also an invariant set.