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Mathematicians have devised different chaotic systems that are
modeled by integer or fractional-order differential equations, and
whose mathematical models can generate chaos or hyperchaos. The
numerical methods to simulate those integer and fractional-order
chaotic systems are quite different and their exactness is
responsible in the evaluation of characteristics like Lyapunov
exponents, Kaplan-Yorke dimension, and entropy. One challenge is
estimating the step-size to run a numerical method. It can be done
analyzing the eigenvalues of self-excited attractors, while for
hidden attractors it is difficult to evaluate the equilibrium
points that are required to formulate the Jacobian matrices. Time
simulation of fractional-order chaotic oscillators also requires
estimating a memory length to achieve exact results, and it is
associated to memories in hardware design. In this manner,
simulating chaotic/hyperchaotic oscillators of
integer/fractional-order and with self-excited/hidden attractors is
quite important to evaluate their Lyapunov exponents, Kaplan-Yorke
dimension and entropy. Further, to improve the dynamics of the
oscillators, their main characteristics can be optimized applying
metaheuristics, which basically consists of varying the values of
the coefficients of a mathematical model. The optimized models can
then be implemented using commercially available amplifiers,
field-programmable analog arrays (FPAA), field-programmable gate
arrays (FPGA), microcontrollers, graphic processing units, and even
using nanometer technology of integrated circuits. The book
describes the application of different numerical methods to
simulate integer/fractional-order chaotic systems. These methods
are used within optimization loops to maximize positive Lyapunov
exponents, Kaplan-Yorke dimension, and entropy. Single and
multi-objective optimization approaches applying metaheuristics are
described, as well as their tuning techniques to generate feasible
solutions that are suitable for electronic implementation. The book
details several applications of chaotic oscillators such as in
random bit/number generators, cryptography, secure communications,
robotics, and Internet of Things.
Mathematicians have devised different chaotic systems that are
modeled by integer or fractional-order differential equations, and
whose mathematical models can generate chaos or hyperchaos. The
numerical methods to simulate those integer and fractional-order
chaotic systems are quite different and their exactness is
responsible in the evaluation of characteristics like Lyapunov
exponents, Kaplan-Yorke dimension, and entropy. One challenge is
estimating the step-size to run a numerical method. It can be done
analyzing the eigenvalues of self-excited attractors, while for
hidden attractors it is difficult to evaluate the equilibrium
points that are required to formulate the Jacobian matrices. Time
simulation of fractional-order chaotic oscillators also requires
estimating a memory length to achieve exact results, and it is
associated to memories in hardware design. In this manner,
simulating chaotic/hyperchaotic oscillators of
integer/fractional-order and with self-excited/hidden attractors is
quite important to evaluate their Lyapunov exponents, Kaplan-Yorke
dimension and entropy. Further, to improve the dynamics of the
oscillators, their main characteristics can be optimized applying
metaheuristics, which basically consists of varying the values of
the coefficients of a mathematical model. The optimized models can
then be implemented using commercially available amplifiers,
field-programmable analog arrays (FPAA), field-programmable gate
arrays (FPGA), microcontrollers, graphic processing units, and even
using nanometer technology of integrated circuits. The book
describes the application of different numerical methods to
simulate integer/fractional-order chaotic systems. These methods
are used within optimization loops to maximize positive Lyapunov
exponents, Kaplan-Yorke dimension, and entropy. Single and
multi-objective optimization approaches applying metaheuristics are
described, as well as their tuning techniques to generate feasible
solutions that are suitable for electronic implementation. The book
details several applications of chaotic oscillators such as in
random bit/number generators, cryptography, secure communications,
robotics, and Internet of Things.
This book offers readers a clear guide to implementing engineering
applications with FPGAs, from the mathematical description to the
hardware synthesis, including discussion of VHDL programming and
co-simulation issues. Coverage includes FPGA realizations such as:
chaos generators that are described from their mathematical models;
artificial neural networks (ANNs) to predict chaotic time series,
for which a discussion of different ANN topologies is included,
with different learning techniques and activation functions; random
number generators (RNGs) that are realized using different chaos
generators, and discussions of their maximum Lyapunov exponent
values and entropies. Finally, optimized chaotic oscillators are
synchronized and realized to implement a secure communication
system that processes black and white and grey-scale images. In
each application, readers will find VHDL programming guidelines and
computer arithmetic issues, along with co-simulation examples with
Active-HDL and Simulink.The whole book provides a practical guide
to implementing a variety of engineering applications from VHDL
programming and co-simulation issues, to FPGA realizations of chaos
generators, ANNs for chaotic time-series prediction, RNGs and
chaotic secure communications for image transmission.
This book offers readers a clear guide to implementing engineering
applications with FPGAs, from the mathematical description to the
hardware synthesis, including discussion of VHDL programming and
co-simulation issues. Coverage includes FPGA realizations such as:
chaos generators that are described from their mathematical models;
artificial neural networks (ANNs) to predict chaotic time series,
for which a discussion of different ANN topologies is included,
with different learning techniques and activation functions; random
number generators (RNGs) that are realized using different chaos
generators, and discussions of their maximum Lyapunov exponent
values and entropies. Finally, optimized chaotic oscillators are
synchronized and realized to implement a secure communication
system that processes black and white and grey-scale images. In
each application, readers will find VHDL programming guidelines and
computer arithmetic issues, along with co-simulation examples with
Active-HDL and Simulink.The whole book provides a practical guide
to implementing a variety of engineering applications from VHDL
programming and co-simulation issues, to FPGA realizations of chaos
generators, ANNs for chaotic time-series prediction, RNGs and
chaotic secure communications for image transmission.
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