Abstract
When a new chaotic oscillator is introduced, it must accomplish characteristics like guaranteeing the existence of a positive Lyapunov exponent and a high Kaplan–Yorke dimension. In some cases, the coefficients of a mathematical model can be varied to increase the values of those characteristics but it is not a trivial task because a very huge number of combinations arise and the required computing time can be unreachable. In this manner, we introduced the optimization of the Kaplan–Yorke dimension of chaotic oscillators by applying metaheuristics, e.g., differential evolution (DE) and particle swarm optimization (PSO) algorithms. We showed the equilibrium points and eigenvalues of three chaotic oscillators that are simulated applying ODE45, and the Kaplan–Yorke dimension was evaluated by Wolf’s method. The chaotic time series of the state variables associated to the highest Kaplan–Yorke dimension provided by DE and PSO are used to encrypt a color image to demonstrate that they are useful in implementing a secure chaotic communication system. Finally, the very low correlation between the chaotic channel and the original color image confirmed the usefulness of optimizing Kaplan–Yorke dimension for cryptographic applications.
Highlights
Chaotic systems are a hot topic of interest for researchers in a wide variety of fields
We show the application of two metaheuristics, namely: Differential evolution (DE) and particle swarm optimization (PSO) algorithms, in order to maximize DKY of three chaotic oscillators
This article showed that the chaotic encryption of an image can be enhanced when using chaotic data from an oscillator that has a high Kaplan–Yorke dimension DKY
Summary
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 DKY , 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. The short term unpredictability of the chaotic flow is demonstrated via the calculation of DKY 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, DKY , etc.
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