In mathematics, the tent map with parameter μ is the real-valued function fμ defined by the name being due to the tent-like shape of the graph of fμ. For the values of the parameter μ within 0 and 2, fμ the unit interval [0, 1] into itself, thus defining a discrete-time dynamical system on it (equivalently, a recurrence relation). In particular, iterating a point x0 in [0, 1] gives rise to a sequence : where μ is a positive real constant. Choosing for instance the parameter μ = 2, the effect of the function fμ may be viewed as the result of the operation of folding the unit interval in two, then stretching the resulting interval [0, 1/2] to get again the interval [0, 1]. Iterating the procedure, any point x0 of the interval assumes new subsequent positions as described above, generating a sequence xn in [0, 1]. The case of the tent map is a non-linear transformation of both the bit shift map and the r = 4 case of the logistic map. The tent map with parameter μ = 2 and the logistic map with parameter r = 4 are topologically conjugate, and thus the behaviours of the two maps are in this sense identical under iteration. 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. If μ is 1 all values of x less than or equal to 1/2 are fixed points of the system. If μ is greater than 1 the system has two fixed points, one at 0, and the other at μ/(μ + 1). Both fixed points are unstable, i.e. a value of x close to either fixed point will move away from it, rather than towards it. For example, when μ is 1.5 there is a fixed point at x = 0.6 (since 1.5(1 − 0.6) = 0.6) but starting at x = 0.61 we get If μ is between 1 and the square root of 2 the system maps a set of intervals between μ − μ2/2 and μ/2 to themselves. This set of intervals is the Julia set of the map – that is, it is the smallest invariant subset of the real line under this map.

About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related courses (4)
BIO-341: Dynamical systems in biology
Life is non-linear. This course introduces dynamical systems as a technique for modelling simple biological processes. The emphasis is on the qualitative and numerical analysis of non-linear dynamical
COM-502: Dynamical system theory for engineers
Linear and nonlinear dynamical systems are found in all fields of science and engineering. After a short review of linear system theory, the class will explain and develop the main tools for the quali
PHYS-460: Nonlinear dynamics, chaos and complex systems
The course provides students with the tools to approach the study of nonlinear systems and chaotic dynamics. Emphasis is given to concrete examples and numerical applications are carried out during th
Show more
Related publications (14)
Related concepts (7)
Dyadic transformation
The dyadic transformation (also known as the dyadic map, bit shift map, 2x mod 1 map, Bernoulli map, doubling map or sawtooth map) is the mapping (i.e., recurrence relation) (where is the set of sequences from ) produced by the rule Equivalently, the dyadic transformation can also be defined as the iterated function map of the piecewise linear function The name bit shift map arises because, if the value of an iterate is written in binary notation, the next iterate is obtained by shifting the binary point one bit to the right, and if the bit to the left of the new binary point is a "one", replacing it with a zero.
Baker's map
In dynamical systems theory, the baker's map is a chaotic map from the unit square into itself. It is named after a kneading operation that bakers apply to dough: the dough is cut in half, and the two halves are stacked on one another, and compressed. The baker's map can be understood as the bilateral shift operator of a bi-infinite two-state lattice model. The baker's map is topologically conjugate to the horseshoe map. In physics, a chain of coupled baker's maps can be used to model deterministic diffusion.
Iterated function
In mathematics, an iterated function is a function X → X (that is, a function from some set X to itself) which is obtained by composing another function f : X → X with itself a certain number of times. The process of repeatedly applying the same function is called iteration. In this process, starting from some initial object, the result of applying a given function is fed again in the function as input, and this process is repeated. For example on the image on the right: with the circle‐shaped symbol of function composition.
Show more

Graph Chatbot

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.