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# Topic: Logistic map

###### In the News (Tue 16 Jul 19)

 Bio 481 - Problem Set 1 Construct the bifurcation diagram for the logistic map with constant predation, using r as the bifurcation parameter. Construct the bifurcation diagram for the logistic map with constant predation, using the predation coefficient as the bifurcation parameter. Since some of the logistic equation with predation has a negative trajectory, the plot is moved to the down and to the right. www-personal.umich.edu /~rgrommel/academic/bio481ps1.html   (2308 words)

 [No title]   (Site not responding. Last check: 2007-10-07) # # & The logistic equation is an iterative map. The equation of the map is # x_next = f * x (1 - x). The independant variable in the logistic map is # the "f" term. astronomy.swin.edu.au /~pbourke/fractals/logistic/logistic_map.py   (435 words)

 Logistic Equation: The parable of the parabola The remarkable feature of the logistic map is in the simplicity of its form (quadratic) and the complexity of its dynamics. The logistic map is the simplest model in population dynamics that incorporates the effects of both birth and death rates. As mentioned earlier, an m period limit cycle of the logistic equation is the fixed point of the m composition of the logistic equation, therefore to determine the stability of m period limit cycle, we evaluate the slope of f chaos.phy.ohiou.edu /~thomas/chaos/logistic.html   (1617 words)

 Logistic map at opensource encyclopedia   (Site not responding. Last check: 2007-10-07) The logistic map is an archetypical example of how very complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was broadly known due to a seminal paper by the biologist Robert May in 1976. The logistic model was originally introduced as a demographic model by Verhulst. www.wiki.tatet.com /Logistic_map.html   (1250 words)

 Stochastic Logistic Map Bifurcations of the Randomly Perturbed Logistic Map in the way that it contains colored versions of all pictures given there and many new numerical results which are partly presented as movies. We are aiming at a thorough study of the stochastic bifurcation behavior of the discrete-time random dynamical system generated by the stochastically perturbed logistic map. This study focuses on the effect of noise on the bifurcations of the logistic map, and it aims at an understanding of the stochastic bifurcation behavior of the randomly perturbed logistic map. www.iew.unizh.ch /home/klaus/logistic/intro.html   (603 words)

 Exploring the Logistic Map   (Site not responding. Last check: 2007-10-07) You will notice that the map is quite smooth for low values of gain and it has some distinct breaks in it. The complicated tangle of the map in this area must be presented on a grid of points with limited resolution. Iterating a function deeply and displaying it on a phase-control map in effect reveals the shape of the attractor in that phase-control space. www.mcasco.com /explorin.html   (1423 words)

 Mathcad Library Unfortunately the logistic map could not be used as an infinite (aperiodic) random number generator because of the strong correlations between the generated sequences of numbers. The logistic map is able to generate an infinite chaotic sequence of numbers. By construction, the sequence of digits (15) generated with the described algorithm is not correlated and aperiodic, because the sequence of numbers generated with the logistic maps (13) is not correlated and aperiodic. www.mathcad.com /Library/LibraryContent/MathML/logrnd.htm   (2041 words)

 Introduction to Nonlinear Dynamics in the Logistic Map Similarly, a periodic point of a map f is a point x such that x = f^n(x) for some positive integer n. For a is between 0 to 1, the map has a single fixed point at 0 which is stable. Note that a consequence of the chain rule is that all points on the periodic trajectory become unstable at the same parameter value. www.geocities.com /chaiwahwu/chaosintro/logistic_map.html   (806 words)

 Dirk Holste / Lomas   (Site not responding. Last check: 2007-10-07) Logistic map for the parameter range A [2.8; 4.0]. The Liapunov exponent of the logistic map is included in the bifurcation diagram allowing to associate the exponent with the bifurcation point. Bifurcation diagram of the logistic map perturbed by Gaussian white noise and the corresponding Liapunov exponent. itb.biologie.hu-berlin.de /~dirk/lomas.html   (196 words)

 Fractal Geometry The logistic map is defined by a parabola, the tent map by a broken line, both symmetric about x = 1/2. All the various dynamics of the tent map and of the logistic map can be assembled into the Tent and Logistic Bifurcation Diagrams. Against the objections that the logistic map is a very special case, we offer the universality of the logistic map dynamics. classes.yale.edu /Fractals/Chaos/welcome.html   (1471 words)

 Miller's Mathematical Ideas, 9th Edition Web site Chapter 9 -- Internet Project   (Site not responding. Last check: 2007-10-07) The logistic function is not some mathematical contrivance designed to bewilder students, it plays a very important role in the study of population growth. The logistic function is used to generate sequences of numbers as follows. Although the logistic map has a very simple definition, a calculator is a necessary tool in the generation of logistic sequences. occawlonline.pearsoned.com /bookbind/pubbooks/miller2_awl/chapter9/essay1/deluxe-content.html   (1053 words)

 Notes Section 12 It is important to distinguish between maps that are noninvertible and maps that are invertible. It may be regarded as of normal form type since iteration of any quadratic polynomial is equivalent to iteration of the logistic map (with properly chosen parameters)... As an example of this aspect compare the bifurcation diagram of the sine map with that of the logistic map. www.sun.rhbnc.ac.uk /~uhap045/316/sect12/sect12.html   (964 words)

 CBofN - Glossary - L The XOR mapping defines two sets of points that are linearly inseparable. Note that this particular valley (or peak) may not necessarily be the lowest (or highest) location in the space, which is referred to as the global minimum (maximum). Logistic Map The simplest chaotic system that works in discrete time and is defined by the map x(t) = 4r x(t) (1-x(t)). mitpress.mit.edu /books/FLAOH/cbnhtml/glossary-L.html   (456 words)

 [No title] In case of the logistic map, this can be archived by using the OPF chaos control method. Controlling the logistic map using the OPF method serves as a basic example to demonstrate important features of chaos control. In practice however, the mapping function of the nonlinear system might not be known and information of the system dynamics must then be extracted from time series data. chaos4.phy.ohiou.edu /~thomas/chaos/chaos_demonstration.html   (910 words)

 [No title]   (Site not responding. Last check: 2007-10-07) Logistic map is the simplest such map which gives non-trivial behavior. Other maps have been proposed as more accurate descriptions of populations density variation over generations. But these "improved" maps have qualitatively the same dynamics as the logistic map. www.imsc.res.in /~sitabhra/research/persistence/logistic_ecology.html   (221 words)

 NDS The Logistic Map prevents unlimited growth by inhibiting growth whenever it achieves a high level. It turns out that the logistic map is a very different animal, depending on its control parameter r. Behavior of the Logistic map for r=1.25, 2.00, and 2.75. www.vanderbilt.edu /AnS/psychology/cogsci/chaos/workshop/NDS.html   (480 words)

 The Logistic Map   (Site not responding. Last check: 2007-10-07) The logistic map is recursive, meaning that the third term is a function of the second, the fourth a function of the third and so on. It is perhaps best to think of the logistic map as nothing more than the equation for an harmonic oscillator. Further graphs of the logistic map will continue to show this close relationship to the harmonic oscillator, and in fact a driven pendulum, a very common object, is known to show chaotic behavior. cannon.sfsu.edu /~mstevens/chaos/chaos.htm   (526 words)

 [No title] If these data come from a logistic map or any one-dimensional map, then the Nth vector will have the form: X[N] = (x[N], f(x[N])), and you should be able to see that all the 2-vectors will collapse on to the quadratic curve (x,f(x)) rather than being sprinkled over the plane. This idea generalizes to higher-dimensional spaces although one needs to turn to a computer algorithm since it is difficult to visualize structures in 3d, 4d, etc spaces. This may seem phony but there is a change of variables: x[i] = (1/2) (1 - Cos(Pi y[i])), which converts the logistic map with r=4 into the tent map for y[i]. www.phy.duke.edu /~hsg/213/lectures/9-8-03.txt   (847 words)

 Bio 481 - Problem Set 5   (Site not responding. Last check: 2007-10-07) Using a chaotic logistic equation try two different starting N0's (in the chaotic regime) and project the population. Look at the logistic map (Nt+1 vs. Nt) at some different r values with chaotic dynamics. Do a bifurcation plot of the logistic map with constant predation (bifurcate with P). www-personal.umich.edu /~rgrommel/academic/bio481ps5.html   (337 words)

 PlanetMath: Feigenbaum fractal   (Site not responding. Last check: 2007-10-07) A Feigenbaum fractal is any bifurcation fractal produced by a period-doubling cascade. The ``canonical'' Feigenbaum fractal is produced by the logistic map (a simple population model), The logistic iteration either terminates in a cycle (set of repeating values) or behaves chaotically. planetmath.org /encyclopedia/LogisticMap.html   (170 words)

 The Logistic Map, Period-Doubling To Chaos Next we discuss a remarkable theorem on the dynamics of real maps (such as the logistic map). This is confirmed in the logistic map case. Question: In the logistic map, why, of all the infinitely many periodic orbits, at most one is attracting? www.emba.uvm.edu /math/classes/math266/notes_6_3.htm   (901 words)

 Numerical Errors in Logistic Map Calculation Suppose that the logistic equation with A = 4 is iterated using (4-byte) single-precision arithmetic. Four bytes is 32 bits, of which 23 are used for the mantissa, 8 for the exponent, and 1 for the sign according to the IEEE standard for floating point arithmetic. To test this prediction, the logistic map was iterated in single precision many times using a uniform random initial condition in (0,1), and the number of iterations required for zero to be reached was determined. sprott.physics.wisc.edu /chaos/nelogmap.htm   (774 words)

 Logistic Map   (Site not responding. Last check: 2007-10-07) The logistic map (and its continuous counterpart, the logistic equation) are well studied models in population biology. Some number streams from the logistic map converted into note streams. The logistic map is used to the model the environments as it can easily be modified between its stable, periodic and chaotic areas. homepages.which.net /~gk.sherman/kc.htm   (127 words)

 Logistic Map.   (Site not responding. Last check: 2007-10-07) Basically, this map, like any one-dimensional map, is a rule for getting a number from a number. The parameter a is fixed, but if one studies the map for diffrent values of a (up to 4, else the unit interval is no longer invariant) it is found that a is the catalyst for chaos. By drawing a straight line to the diagonal from the graph, you are in fact "resetting" the plot with an inital condition that is the result of your previous inital condition. math.la.asu.edu /~chaos/logistic.html   (241 words)

 Chaos Theory The formula for the logistic map is f(x) = r * x * (1 - x), where r is a constant. behavior of the logistic map depends greatly on the value of r. If you make a graph with r as the x-axis going from 0 to 4, and the value the logistic map settles to as the y-axis, you get what is known as the bifurcation tree. alumni.imsa.edu /~stendahl/chaos/chaos.html   (700 words)

 ipedia.com: Dynamical system Article   (Site not responding. Last check: 2007-10-07) In engineering and mathematics, a dynamical system is a deterministic process in which a function's value changes over time according to a rule that is defined in terms of the function's current value. The logistic map is only a second-degree polynomial; the horseshoe map is piecewise linear. Horseshoe map is an example of a chaotic piecewise linear map www.ipedia.com /dynamical_system.html   (532 words)

 Introduction This exhibited similar behaviour to the usual logistic map (a=b=1 in (2)). is unity, whereas for the Stutzer map this slope is zero. The simplest extension of this map to the case of both exponents fractional (with a > 1) is to have a=3/2 and b=1/2, and this will now be analysed in detail. journal-ci.csse.monash.edu.au /ci/vol02/gottlieb/node1.html   (343 words)

 Logistic Map The logistic map (below) is a graph for all values of g from 0 to 4 that shows the end result(s) of the equation. The population growth (g) is on the x (horizontal) axis, and the final populations are shown on the y (vertical) axis. To display a graph, enter the growth and initial population or click a place on the logistic map and hit Graph. www.geocities.com /capecanaveral/hangar/7959/logisticmap.html   (453 words)

 Chaos and fractals The left panel shows the graph of the Tent Map with the default value of s (1.5) in the Param box. Compare the histogram of the Logistic Map for s = 3.999 with that of the Tent Map for s = 1.999. The 2-cycles of the Tent Map are never stable for any s, while those of the Logistic Map are for some values of s. www.math.union.edu /research/chaos/DynamicsLab.html   (1011 words)

 Deterministic Logistic Map   (Site not responding. Last check: 2007-10-07) The deterministic logistic map is the well known difference equation This deterministic equation can be obtained from the stochastic logistic map by choosing (the ergodic processes) Bifurcation diagram for the second iterate of the logistic map www.iew.unizh.ch /home/klaus/logistic/detlog.html   (132 words)

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