Electronically-implemented coupled logistic maps

Abstract

The logistic map is a paradigmatic dynamical system originally conceived to model the discretetime demographic growth of a population, which shockingly, shows that discrete chaos can emerge from trivial low-dimensional non-linear dynamics. In this work, we design and characterize a simple, low-cost, easy-to-handle, electronic implementation of the logistic map. In particular, our implementation allows for straightforward circuit-modifications to behave as different one-dimensional discrete-time systems. Also, we design a coupling block in order to address the behavior of two coupled maps, although, our design is unrestricted to the discrete-time system implementation and it can be generalized to handle coupling between many dynamical systems, as in a complex system. Our findings show that the isolated and coupled maps' behavior has a remarkable agreement between the experiments and the simulations, even when fine-tuning the parameters with a resolution of ∼ 10 −3 . We support these conclusions by comparing the Lyapunov exponents, periodicity of the orbits, and phase portraits of the numerical and experimental data for a wide range of coupling strengths and map's parameters.PACS. 0 5.45.-a -0 7.05.Fb -0 7.50.Ek

Parameter space of experimental chaotic circuits with high-precision control parameters

“…In particular, we showed by calculating the Lyapunov exponents numerically and experimentally that this parameter space presents self-similar periodic structures, the shrimps, embedded in a domain of chaos. [1][2][3][4][5][6][7][8]10,11,13,14 We also show experimentally that those self-similar periodic regions organize themselves in period-adding bifurcation cascades, and whose sizes decrease exponentially as their period grows. 1,9,17,18 We also report on malformed shrimps on the experimental parameter space, result of tiny nonlinear deviations close to the junction of two linear parts from a symmetric piecewise linear i(V) curve.…”

“…Only a few chaotic circuits have periodicity high resolution parameter spaces experimentally obtained. [1][2][3][4][5][6][7] The relevance of studying parameter spaces of nonlinear systems is that it allows us to understand how periodic behavior, chaos, and bifurcations come about in a nonlinear system. In fact, parameters leading to the different behaviors are strongly correlated.…”

Observation of bifurcations and hysteresis in experimentally coupled logistic maps

“…1) up to an error less than 3% with respect to the theoretical quadratic function. Its discrete-time evolution is implemented by a sample-andhold circuit [28], which holds the voltage output, V out , constant for a fixed time-window, before releasing it to our coupling block circuit; details in A The resultant configurations are adaptable, as shown in the bottom panels of Fig. 1.…”

“…Our experimental set-up is based on the logistic map circuit we define in Ref. [28], which we now extend to include a circuit board that allows to change the coupling configuration (i.e., the connectivity between maps) as well as the number of interacting maps. The circuit allows precise and reliable manipulations (with an average 1% parameter uncertainty per map) with high signal-to-noise ratio (∼ 10 6 ).…”

A New Simple Chaotic Lorenz-Type System and Its Digital Realization Using a TFT Touch-Screen Display Embedded System

“…Since then, many chaotic systems like Lorenz-and their chaotic behavior-have been reported in the literature, for example, [17][18][19][20][21][22][23]. Currently, we can mention some new chaotic systems reported in the literature [24][25][26][27][28][29][30][31][32][33].…”