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Chaotic, Forced, and Coupled Oscillators
Published in LM Pismen, Working with Dynamical Systems, 2020
What makes the logistic map important is that it reproduces the period-doubling route to chaos observed in many dynamical systems, and provides much easier means to explore its details. The first stages of the period-doubling cascade, taking place as the parameter μ increases, are illustrated in Fig. 4.7. At small values of μ, the logistic map has, besides the unstable trivial equilibrium, a single fixed point x=x0, which is stable as long as the slope f′(x) is less than one by its absolute value. At μ =3, a flip bifurcation takes place: above it, f′(x0) < −1, so that the fixed point is unstable, and the system approaches a period-two sequence, alternating between two values x = x1± upon each iteration (Fig. 4.7a). Both points are the stable fixed points of the composite map f1(x) = f(f (x)), while x0 is an unstable fixed point of this map with f1′ (x0)> 1 (Fig. 4.7b). As μ further grows, |f1′(x1±)| increases, and a new period doubling takes place at μ ≈3.4495. Above this point, a period-four orbit emerges, where the four alternating values of x are stable fixed points of the composite map f2(x) = f1(f1(x)) = f4(x) (Fig. 4.7c).
A Chaotic Logic-Based Physical Unclonable Computing System
Published in Mark Tehranipoor, Domenic Forte, Garrett S. Rose, Swarup Bhunia, Security Opportunities in Nano Devices and Emerging Technologies, 2017
Cory Merkel, Mesbah Uddin, Garrett S. Rose
As an example, a simple function that has been shown to exhibit chaotic behavior is the logistic map. The logistic map is the recurrence relation associated with the logistic function, often used to model population growth. The logistic map is typically written as follows: xn+1=rxn(1−xn), where r is a positive real number and xn is the state variable of this particular system. When the logistic map is used to represent population dynamics, r represents the combined rate for reproduction and starvation. The state variable xn represents the ratio of the existing population to the maximum possible population for iteration n.
Fractals and chaos
Published in A. W. Jayawardena, Environmental and Hydrological Systems Modelling, 2013
Bifurcation generally means separation of a structure into two branches; however, in the context of dynamical systems, it refers to period doubling, quadrupling, etc., that lead to the onset of chaos. The bifurcation diagram for the logistic map, which is a fractal, is a plot of the values of the parameter r along the horizontal axis and the possible long-term values of x along the vertical axis (Figure 10.8). The bifurcation diagram shows that period doubling bifurcation leads to a new behaviour with twice the period of the original system and period halving bifurcation leads to a new behaviour with half the period of the original system. Period halving leads from chaos to order, whereas period doubling leads from order to chaos. Logistic map bifurcation diagram is a clear illustration of the link between fractals and chaos. The points of separation of the bifurcation diagram of the logistic map looks like a Cantor set with gaps of varying sizes.
Advanced 5D logistic and DNA encoding for medical images
Published in The Imaging Science Journal, 2023
Bharti Ahuja, Rajesh Doriya, Sharad Salunke, Md. Farukh Hashmi, Aditya Gupta
Figure 1 depicts the u vs. y graph of equation (1). In this equation, y represents the population at any given time n, and u represents the rate of growth. Logistic Map is used as a model or a data source for many diversified applications such as life sciences, chemistry, computer vision, etc. Many notable successes in biology have already been achieved by many biologists if they could predict the growth rate of living things so precisely that population size fluctuates between preset values over a timeframe of 2, 4, or 8 years. Also, it may find applications where it is used to model specific chemical processes, such as those described by Malek and Gobal [34]. Because of its essential qualities, such as responsiveness to the starting condition or controlling parameters, the logistic map is widely used for creating novel cryptographic systems to suit the needs of secure image transfer also [35].