Using the logistic map to generate scratching sounds the. In order to expand chaotic region of logistic map and make it suitable for cryptography, two modified \ versions of logistic map are proposed. Properties of invariant distributions and lyapunov. Finite lyapunov exponent for generalized logistic maps. A nonlinearly modulated logistic map with delay for image. Estimating the largest lyapunov exponent based on conditional. Logistic map as a fourier s series chaos based cryptography. Chaos, bifurcation diagrams and lyapunov exponents with r. Numerical calculation of the lyapunov exponent for the. Experimental data inevitably contain external noise due to environmental fluctuations and limited. Scaling relations in the lyapunov exponents of one. Usually only the largest of them is called lyapunov exponent, or more accurately the maximal lyapunov exponent mle. Determining lyapunov exponents from a time series in ref.
Firstly i will concentrate on measuring the recurrence relation. Lyapunov exponents for the logistic map from the wolfram demonstrations projecta wolfram web resource. Onedimensional 1d chaotic maps, such as logistic map 12, usually have relatively narrow chaotic range, smaller lyapunov exponent, and excessive periodic windows. If the velocity of the fluid is not very large the fluid flows in a smooth steady way, called laminar flow, which can be calculated for simple geometries. Press plc chaos and graphics lyapunov exponents of the logistic map with periodic forcing mario markusi and. The resulting images have aesthetically appealing selfsimilar. Chapter 11 stability and lyapunov exponents recall the analysis of stability for the xed point of a map. This demonstration shows a finite lyapunov exponent of a onedimensional unimodal map, which is a generalization of the wellknown logistic map. The points x0 and are both fixed points of the map. The map was popularized in a 1976 paper by the biologist robert may, in part as a discretetime demographic model analogous to.
Properties of invariant distributions and lyapunov exponents. Logistic growth, s curves, bifurcations, and lyapunov. The object shows the crucial belief of the deterministic chaos theory that brings a new procedural structure and apparatus for exploring and understanding complex behavior in dynamical systems. Calculation lyapunov exponents for ode file exchange. Estimating the lyapunov exponents of chaotic time series.
As can be seen in the above plot, a bifurcation in the red map is indicated when the lyapunov exponent blue approches zero green line. Request pdf numerical calculation of the lyapunov exponent for the logistic map chaotic maps can be used to describe the behavior of dynamical systems and. The logistic map introduction one of the most challenging topics in science is the study of chaos. The assessment of applying chaos theory for daily traffic. In this page, the lyapunov exponent is applied to an equation that jumps between stability and instability, between chaos and order the logistic equation. Modification of the logistic map using fuzzy numbers with application to pseudorandom number generation and image encryption subject. Unlike the linear map, calculating the lyapunov exponent is not trivial since the derivative is not constant for the m. The logistic map is obtained by replacing the logisi will then study two aspects of these chaotic sys tic equation for population growth with a quadratic tems. Jul 22, 2014 calculating the lyapunov exponent of a time series with python code posted on july 22, 2014 by neel in a later post i discuss a cleaner way to calculate the lyapunov exponent for maps and particularly the logistic map, along with mathematica code. This demonstration plots the orbit diagram of the logistic map and the corresponding lyapunov exponents for different ranges of the parameter the lyapunov exponent is. Lyapunov exponents of the logistic map with periodic. Lyapunov exponents for multiparameter tent and logistic. R can be used to get the flavor of this richness and reproduce some of the most famous pictures in the history of science, such as the bifurcation diagram of the logistic map or the representation of its lyapunov exponents.
The lyapunov exponent is a number that measures stability. The logistic map is one of the most important but common examples of chaotic dynamics. This may be done through the eigenvalues of the jacobian matrix j 0 x 0. The map was popularized in a 1976 paper by the biologist robert may, in part as a discretetime demographic model analogous to the logistic equation first. Oct 11, 2011 the behaviour of logistic and tent maps is studied in cases where the control parameter is dependent on iteration number. Onedimensional 1d chaotic maps, such as logistic map 12, usually have relatively narrow chaotic range, smaller lyapunov exponent, and excessive periodic windows, and their structure and chaotic orbit are rather simple. Iteration of onedimensional maps can generate stunning complexity and famed examples of chaotic behavior. For continuoustime models, the iterated function is replaced. Vastano, determining lyapunov exponents from a time series, physica d, vol. Observation of different behaviors of logistic map for. Roussel november 15, 2005 in our previous set of notes, we examined the connections between differential equations and maps.
The lyapunov exponent a measure of average stability is displayed with high resolution on the abplane. The derivative can be evaluated by the chain rule in terms of derivatives of fat the intermediate iterations. The rst is the logistic map, a rstorder discrete dynamical system, and the second is the lorenz system, a threedimensional system of di erential. We have studied the bifurcation structure of the logistic map with a time dependant control parameter.
Lyapunov exponents and strange attractors in discrete and continuous dynamical systems jo bovy jo. Bifurcation structure and lyapunov exponents of a modulated logistic map k p harikrishnan and v m nandakumaran department of physics, cochin university of science and technology, cochin 682 022, india ms received 3 june 1987. The obtained results confirm that the analyzed model can safely and effectively replace a classic logistic map for applications involving chaotic cryptography. Finite lyapunov exponent for generalized logistic maps with z. Mar 18, 2004 lyapunov exponent calcullation for odesystem.
A bifurcation diagram visualizes the appearance of period doubling and chaotic behavior as a function of a control parameter. We observe a symmetry of lyapunov exponents in bifurcation structures of onedimensional maps in which there exists a pair of parameter values in a dynamical system such that two dynamical systems with these paired parameter. The results were good approximations of those found in the literature, where the same parameters were used for comparison, with showed in table 1. In a later post i discuss a cleaner way to calculate the lyapunov exponent for maps and particularly the logistic map, along with mathematica code. Scaling relations in the lyapunov exponents of one dimensional maps vol. In the next step, an analysis of lyapunov exponent and the distribution of the iterative variable are studied. Lyapunov exponents of the logistic map with periodic forcing. As you adjust the growth rate parameter upwards, the logistic map will oscillate between two then four then eight then 16 then 32 and on and on population. Development of a lyapunov exponent based chaos diagram in the parameter plane of logistic map. We put an importance on report of the verhulst logistic map which is one of the potential. Development of a lyapunov exponent based chaos diagram in the. The logistic map is a polynomial mapping equivalently, recurrence relation of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple nonlinear dynamical equations. Jul 25, 2015 if youve ever wondered how logistic population growth the verhulst model, s curves, the logistic map, bifurcation diagrams, sensitive dependence on initial conditions, orbits, deterministic chaos, and lyapunov exponents are related to one another this post explains it in just 10 steps, each with some code in r so you can explore it all yourself.
Bifurcation structure and lyapunov exponents of a modulated. Symmetry of lyapunov exponents in bifurcation structures. Modified logistic maps for cryptographic application. If the lyapunov exponent is positive, the system is chaotic. The neural network has superior performance for short periods with length down to 10 lyapunov times on which the traditional lyapunov exponent computation is far from converging. As it so often goes with easy ideas, it turns out that lyapunov exponents are not natural for study of dynamics, and we would have passed them. Lyapunov exponents and strange attractors in discrete and. Us9116838b2 determining lyapunov exponents of a chaotic.
The lyapunov exponent a measure of average stability is displayed with high resolution on the a5plane. In order to expand chaotic region of logistic map and make it suitable for cryptography, two. Lyapunov exponents for multiparameter tent and logistic maps. The resulting map is analysed through its lyapunov exponent le and bifurcation diagrams. The phasemodulated logistic map follows a similar iteration equation, but due to the dependance on. Lyapunov exponent of logistic map file exchange matlab. The number of iterations for an estimate of the lyapunov exponent of the logistic map using the proposed method is similar to the numbers obtained by rosenstein 12 logistic map, as seen in table 2. Chaos, bifurcation diagrams and lyapunov exponents with r 2. Lyapunov exponent of the logistic map mathematica code. Using the logistic map to generate scratching sounds. Lyapunov exponent calculation is done numerically using the standard formulation. Analytic results for global lyapunov exponent are presented in the case of the tent map and numerical results are presented in. In this quick tutorial, ill show you a cleaner way to get the lyapunov exponent for the specific case of the logistic map, and then using a really short script in mathematica, plot it. Development of a lyapunov exponent based chaos diagram in.
We have studied the bifurcation structure oc the logistic map with a time dependant. I actually have go through and that i am sure that i will planning to read once again again in the future. Request pdf numerical calculation of the lyapunov exponent for the logistic map chaotic maps can be used to describe the behavior of dynamical systems and they are characterized by a parameter. Ajide 1 1 department of mechanical engineering, university of ibadan, nigeria. In a previous post id shown a way to get the lyapunov exponent from the time series data of any map. Whereas the global lyapunov exponent gives a measure for the total predictability of a system, it is sometimes of interest to estimate the local predictability around a point x 0 in phase space. If the lyapunov exponent is negative, we typically have. Unlike the linear map, calculating the lyapunov exponent is not trivial since the derivative is not constant for the map. Chapter 4 one dimensional maps california institute of. Dec 12, 2016 the first part of this article can be read hereiteration of onedimensional maps can generate stunning complexity and famed examples of chaotic behavior. By introducing a specific nonlinear variation for the parameter, we show that the bifurcation structure is modified qualitatively as well as quantitatively from the first bifurcation onwards.
May 15, 2015 in a previous post id shown a way to get the lyapunov exponent from the time series data of any map. This paper serves as an introduction to the analysis of chaotic systems, with techniques being developed by working through two famous examples. The results were good approximations of those found in the. The behaviour of logistic and tent maps is studied in cases where the control parameter is dependent on iteration number. A periodic point can only bifurcate if its lyapunov exponent is zero. As an example of chaos, consider fluid flowing round an object. Analytic results for global lyapunov exponent are presented in the case of the tent map and numerical results are presented in the case of the logistic map. In this paper, definition and properties of logistic map along with orbit and bifurcation diagrams, lyapunov exponent, and its histogram are considered. Calculating the lyapunov exponent of a time series with. Maps also arise directly in certain applications, so we have good reason to understand their behavior. The alogrithm employed in this mfile for determining lyapunov exponents was proposed in a. We can, however, do a few computational experiments to better understand this map. If youve ever wondered how logistic population growth the verhulst model, s curves, the logistic map, bifurcation diagrams, sensitive dependence on initial conditions, orbits, deterministic chaos, and lyapunov exponents are related to one another this post explains it in just 10 steps, each with some code in r so you can explore it all yourself. We have also computed the two lyapunov exponents of the system and.
32 1281 669 422 54 958 1150 1307 1516 270 642 823 1067 526 1210 997 1442 230 793 1493 697 62 399 1284 479 73 844 1134 1343 1307 1235 218