logistic map


August 8, 2022

A logistic map is a discrete dynamical system defined by a quadratic difference equation (recurrence formula) xn+1 axn(1 − xn). Also called logistic map or discrete logistic equation. It is known for producing surprisingly complex behavior from simple quadratic formulas. In the logistic map, a is a constant called a parameter, and x is a variable. If you decide the value of a appropriately, decide the first x0, and calculate, you will get a series of numbers x0, x1, x2, …. This sequence is called an orbital in the field of dynamical systems, and the orbital changes depending on what value is given to a. When the parameter a is changed, the trajectory of the logistic map changes in various ways, such as settling to one value, repeating some values ​​periodically, and exhibiting aperiodic fluctuations called chaos. From the standpoint of looking at the logistic map as a model that represents the population of organisms, the variable xn means the number of individuals expressed in each generation, such as the first generation, the second generation, and so on. This formula calculates the number of individuals xn+1 in the next generation from the number xn. The logistic map as an organism population model assumes a situation in which the population of a certain organism lives in a certain environment, and there is no movement of individuals between the environment and the outside, and xn is an accurate means the ratio to the maximum number of individuals that can exist in the environment, not the number of individuals themselves. The logistic map can also be derived from the discretization of the logistic equation, which models the number of individuals using differential equations, hence the name "logistic map". Although there have been studies of quadratic functions as dynamical systems since the early 20th century, the logistic map became widely known in the 1970s, especially due to the work of the mathematical biologist Robert May. Besides May, Stanislaw Ulam and John von Neumann, Pekka Muhlberg, Oleksandr Szarkowski, Nicholas Metropolis, and Mitchell Feigenbaum have done work on the behavior of the logistic map. .

Definition and Background

Simply put, the logistic map is the following quadratic function. The a in the formula means a constant and is called a parameter. In addition to a, μ, r, and λ are sometimes used as parameter symbols. For formula (1-1), first determine the specific value of the constant a. Furthermore, determine the value of the variable x appropriately and calculate the value of y from the formula. Then, the obtained y value is regarded as the new x value and substituted into the formula to calculate the new y value. Repeating such calculations is what is done using the logistic map. As an example, when a is 2 and the first x is 0.01, repeating the calculation up to the 5th time results in the following table. The logistic map itself is a simple quadratic function that you learn in junior high school, and the calculation itself is possible even for junior high school students. Repetitive calculations can be done with a calculator, and can be made easier with computer spreadsheet software. The logistic map can also be written in the form of a recurrence equation or a difference equation as follows. Once the value of x0 is determined, the values ​​of subsequent variables x1, x2, x3,... can be calculated sequentially according to this difference equation. A difference equation that produces such a sequence is called a dynamical system (D