Chaotic systems are a hot topic of interest for researchers in a wide variety of fields. In the case of optimization and applications, one is interested in finding appropriate design parameters that provide better characteristics like a high positive Lyapunov exponent (LE+) and a high Kaplan–Yorke dimension , and guaranteeing chaotic behavior for long-times. This task can be performed by varying the coefficients of a mathematical model under ranges that can be established from the evaluation of the bifurcation diagram. However, this requires extensive computing time and is more complex when the values of the coefficients have more fractional values. For example: The Lorenz chaotic oscillator has three design parameters: , and . They can have integer values and also fractional ones. If one chose varying them using two integer () and four fractional numbers (), then the number of combinations becomes . Simulating this number of cases can be unreachable in a couple of years and not all cases will generate chaotic behavior. In this manner, metaheuristics can be applied to search for the best coefficient values that provide high LE+ and high .
Among all the different kinds of chaotic systems, the case studies of this work are three dimensional autonomous chaotic oscillators, and the analysis to determine their equilibrium points and eigenvalues are shown. The numerical simulation is performed by applying ODE45 to generate chaotic time series that are used to evaluate both LE+ and
by applying Wolf’s method [
1]. In fact, chaotic flow is interesting for systems that have high complexity that can be quantified by evaluating the attractor dimension, which is associated to
. For instance, the authors in [
2] introduced a flexible chaotic system through applying modification in a recent rare chaotic flow which has adjustable
, and it is used in a practical application showing a relation between
and the ability of the chaotic system to generate random numbers. The authors in [
3] published a review paper of fully analog realizations of chaotic dynamics that can be considered canonical (minimum number of the circuit elements), robust (exhibit structurally stable strange attractors), and novel. The short term unpredictability of the chaotic flow is demonstrated via the calculation of
that is high, so that the generated chaotic waveforms can find interesting applications in the fields of chaotic masking, modulation, or chaos-based cryptography. Another new chaotic oscillator is proposed in [
4], where the authors present a systematic study including phase portraits, dissipativity, stability,
, etc. In the same line of research, the authors in [
5] analyze a chaotic satellite system using dissipativity, equilibrium points, bifurcation diagrams, Poincaré section maps, and
to ensure the strange behavior of the chaotic system. As one can infer,
is quite useful to characterize a chaotic dynamical system that can be implemented with electronics for engineering applications like secure chaotic communication systems and it can also be useful to model natural dynamics like the predator-prey system given in [
6]. In this manner, we show the application of two metaheuristics, namely: Differential evolution (DE) and particle swarm optimization (PSO) algorithms, in order to maximize
of three chaotic oscillators. The state variables of each chaotic oscillator with the highest
are used to encrypt a color image to demonstrate their usefulness in implementing a chaotic secure communication system.
The rest of the manuscript is organized as follows:
Section 2 describes the three chaotic oscillators that are used to maximize
. They are a chaotic system with infinite equilibria points [
7], Rössler [
8], and Lorenz [
9] systems.
Section 3 details the DE and PSO algorithms that are used to maximize
.
Section 4 details the maximization of
for the three chaotic oscillators and shows statistical results of 10 runs applying DE and PSO.
Section 5 shows the chaotic time series with the highest
that are used to encrypt a color image. Finally, the conclusions are given in
Section 6.