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The logistic map is a polynomial mapping, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popularized in a seminal 1976 paper by the biologist Robert May. The logistic model was originally introduced as a demographic model by Pierre François Verhulst. Later it was applied on surplus production of the population biomass of species in the presence of limiting factors such as food supply or disease, and so two causal effects: In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...
A plot of the Lorenz attractor for values r = 28, σ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under specific conditions exhibit dynamics that are sensitive to initial conditions (popularly referred to as the butterfly effect). ...
To do: 20th century mathematics chaos theory, fractals Lyapunov stability and non-linear control systems non-linear video editing See also: Aleksandr Mikhailovich Lyapunov Dynamical system External links http://www. ...
Year 1976 (MCMLXXVI) was a leap year starting on Thursday (link will display full calendar) of the 1976 Gregorian calendar. ...
Robert McCredie Bob May, Baron May of Oxford OM AC Kt (born 8 January 1936 in Australia) is a cross-bench member of the British House of Lords and President of the Royal Society. ...
Map of countries by population Population growth showing projections for later this century Demography is the statistical study of human populations. ...
An abstract model (or conceptual model) is a theoretical construct that represents something, with a set of variables and a set of logical and quantitative relationships between them. ...
Pierre François Verhulst (October 28, 1804 - February 15, 1849, Brussels, Belgium) was a mathematician and a doctor in number theory from the University of Ghent in 1825. ...
Switchgrass, a hardy plant used in the biofuel industry in the United States Rice chaff. ...
In biology, a species is one of the basic units of biodiversity. ...
In biology, agricultural science, physiology, and ecology, a limiting factor is one that controls a process, such as organism growth or species population size or distribution. ...
- reproduction means the population will increase at a rate proportional to the current population
- starvation means the population will decrease at a rate proportional to the value obtained by taking the theoretical "carrying capacity" of the environment less the current population.
Mathematically this can be written as In mathematics, two quantities are called proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio. ...
where:
- xn is a number between zero and one, and represents the population at year n, and hence x0 represents the initial population (at year 0)
- r is a positive number, and represents a combined rate for reproduction and starvation.
Behaviour dependent on r By varying the parameter r, the following behaviour is observed: - With r between 0 and 1, the population will eventually die, independent of the initial population.
- With r between 1 and 2, the population will quickly stabilize on the value
 , independent of the initial population. - With r between 2 and 3, the population will also eventually stabilize on the same value
- , but first oscillates around that value for some time. The rate of convergence is linear, except for r=3, when it is dramatically slow, less than linear.
- With r between 3 and 1+√6 (approximately 3.45), the population may oscillate between two values forever. These two values are dependent on r.
- With r between 3.45 and 3.54 (approximately), the population may oscillate between four values forever.
- With r slightly bigger than 3.54, the population will probably oscillate between 8 values, then 16, 32, etc. The lengths of the parameter intervals which yield the same number of oscillations decrease rapidly; the ratio between the lengths of two successive such bifurcation intervals approaches the Feigenbaum constant δ = 4.669
 . This behavior is an example of a period-doubling cascade. - At r approximately 3.57 is the onset of chaos, at the end of the period-doubling cascade. We can no longer see any oscillations. Slight variations in the initial population yield dramatically different results over time, a prime characteristic of chaos.
- Most values beyond 3.57 exhibit chaotic behaviour, but there are still certain isolated values of r that appear to show non-chaotic behavior; these are sometimes called islands of stability. For instance, beginning at 1+√8 (approximately 3.83) there is a range of parameters r which show oscillation between three values, and for slightly higher values of r oscillation between 6 values, then 12 etc. There are other ranges which yield oscillation between 5 values etc.; all oscillation periods do occur.
- Beyond r = 4, the values eventually leave the interval [0,1] and diverge for almost all initial values.
A bifurcation diagram summarizes this. The horizontal axis shows the values of the parameter r while the vertical axis shows the possible long-term values of x.
In numerical analysis (a branch of mathematics), the speed at which a convergent sequence approaches its limit is called the rate of convergence. ...
There are two mathematical constants called Feigenbaum constants, named after mathematician Mitchell Feigenbaum. ...
A Period doubling bifurcation in a dynamical system is a bifurcation in which the system switches to a new behavior with twice the period of the original system. ...
In mathematics, particularly in dynamical systems, a bifurcation diagram shows the possible long-term values a variable of a system can obtain in function of a parameter of the system. ...
 Bifurcation diagram for the Logistic map The bifurcation diagram is a fractal: if you zoom in on the above mentioned value r = 3.82 and focus on one arm of the three, the situation nearby looks like a shrunk and slightly distorted version of the whole diagram. The same is true for all other non-chaotic points. This is an example of the deep and ubiquitous connection between chaos and fractals. Image File history File links Download high resolution version (1838x1300, 570 KB) Summary A bifurcation diagram for the Logistic map: The horizontal axis is the r parameter, the vertical axis is the x variable. ...
Image File history File links Download high resolution version (1838x1300, 570 KB) Summary A bifurcation diagram for the Logistic map: The horizontal axis is the r parameter, the vertical axis is the x variable. ...
The boundary of the Mandelbrot set is a famous example of a fractal. ...
A GNU Octave script to generate bifurcation diagrams
is available. For other uses of the word octave see Octave (disambiguation) Octave is a free computer program for performing numerical computations, which is mostly compatible with MATLAB. It is part of the GNU project. ...
Image File history File links Burification diagram of a logistic map Released by the author (User:Ap) into the public domain. ...
Chaos and the logistic map The relative simplicity of the logistic map makes it an excellent point of entry into a consideration of the concept of chaos. A rough description of chaos is that chaotic systems exhibit a great sensitivity to initial conditions -- a property of the logistic map for most values of r between about 3.57 and 4 (as noted above). A common source of such sensitivity to initial conditions is that the map represents a repeated folding and stretching of the space on which it is defined. In the case of the logistic map, the quadratic difference equation (1) describing it may be thought of as a stretching-and-folding operation on the interval (0,1). f(x) = x2 - x - 2 In mathematics, a quadratic function is a polynomial function of the form , where a is nonzero. ...
In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...
 Two- and three-dimensional phase diagrams show the stretching-and-folding structure of the logistic map. The following figure illustrates the stretching and folding over a sequence of iterates of the map. Figure (a), left, gives a two-dimensional phase diagram of the logistic map for r=4, and clearly shows the quadratic curve of the difference equation (1). However, we can embed the same sequence in a three-dimensional phase space, in order to investigate the deeper structure of the map. Figure (b), right, demonstrates this, showing how initially nearby points begin to diverge, particularly in those regions of Xt corresponding to the steeper sections of the plot. Image File history File links This image shows scatterplots (in this case, phase diagrams) for the chaotic logistic map with parameter r = 4. ...
Image File history File links This image shows scatterplots (in this case, phase diagrams) for the chaotic logistic map with parameter r = 4. ...
For other senses of this term, see phase space (disambiguation). ...
In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ...
This stretching-and-folding does not just produce a gradual divergence of the sequences of iterates, but an exponential divergence (see Lyapunov exponents), evidenced also by the complexity and unpredictability of the chaotic logistic map. In fact, exponential divergence of sequences of iterates explains the connection between chaos and unpredictability: a small error in the supposed initial state of the system will tend to correspond to a large error later in its evolution. Hence, predictions about future states become progressively (indeed, exponentially) worse when there are even very small errors in our knowledge of the initial state. In mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. ...
Complexity in general usage is the opposite of simplicity. ...
Predictability refers to the degree that a correct prediction of a systems state can be made either qualitatively or quantitatively. ...
In mathematics, exponential growth (or geometric growth) occurs when the growth rate of a function is always proportional to the functions current size. ...
Since the map is confined to an interval on the real number line, its dimension is less than or equal to unity. Numerical estimates yield a correlation dimension of 0.500 ± 0.005 (Grassberger, 1983), a Hausdorff dimension of about 0.538 (Grassberger 1981), and an information dimension of 0.5170976... (Grassberger 1983) for r=3.5699456... (onset of chaos). Note: It can be shown that the correlation dimension is certainly between 0.4926 and 0.5024. In chaos theory the correlation dimension (denoted by ν) is a measure of the dimensionality of the space occupied by a set of random points. ...
In mathematics, the Hausdorff dimension is an extended non-negative real number associated to any metric space. ...
In fractal geometry, the fractal dimension is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. ...
It is often possible, however, to make precise and accurate statements about the likelihood of a future state in a chaotic system. If a (possibly chaotic) dynamical system has an attractor, then there exists a probability measure that gives the long-run proportion of time spent by the system in the various regions of the attractor. In the case of the logistic map with parameter r = 4 and an initial state in (0,1), the attractor is also the interval (0,1) and the probability measure corresponds to the beta distribution with parameters a = 0.5 and b = 0.5. Unpredictability is not randomness, but in some circumstances looks very much like it. Hence, and fortunately, even if we know very little about the initial state of the logistic map (or some other chaotic system), we can still say something about the distribution of states a long time into the future, and use this knowledge to inform decisions based on the state of the system. FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ...
The Lorenz attractor is an example of a non-linear dynamical system. ...
In dynamical systems, an attractor is a set to which the system evolves after a long enough time. ...
In mathematics, a probability space is a set S, together with a σ-algebra X on S and a measure P on that σ-algebra such that P(S) = 1. ...
In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function (pdf) defined on the interval [0, 1]: where α and β are parameters that must be greater than zero and B is the beta function. ...
Decision theory is an interdisciplinary area of study, related to and of interest to practitioners in mathematics, statistics, economics, philosophy, management and psychology. ...
See also The Malthusian growth model, sometimes called the simple exponential growth model, is essentially exponential growth based on a constant rate of compound interest. ...
A plot of the Lorenz attractor for values r = 28, σ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under specific conditions exhibit dynamics that are sensitive to initial conditions (popularly referred to as the butterfly effect). ...
In mathematics, a chaotic map is a map that exhibits some sort of chaotic behavior. ...
Logistic curve, specifically the sigmoid function A logistic function or logistic curve models the S-curve of growth of some set P. The initial stage of growth is approximately exponential; then, as competition arises, the growth slows, and at maturity, growth stops. ...
A radial basis function network is an artificial neural network which uses radial basis functions as activation functions. ...
References
Textbooks - Sprott, Julien Clinton (2003). Chaos and Time-Series Analysis. Oxford University Press. ISBN 0-19-850840-9.
- Strogatz, Steven (2000). Nonlinear Dynamics and Chaos. Perseus Publishing. ISBN 0-7382-0453-6.
- Tufillaro, Nicholas; Tyler Abbott, Jeremiah Reilly (1992). An experimental approach to nonlinear dynamics and chaos. Addison-Wesley New York. ISBN 0-201-55441-0.
Journal Articles A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ...
A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ...
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