Minimal Realizations of Autonomous Chaotic Oscillators Based on Trans-Immittance Filters

Abstract

This review paper describes a design process toward 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. Each autonomous chaotic lumped circuit proposed in this paper can be understood as a looped system, where linear trans-immittance frequency filter interacts with an active nonlinear two-port. The existence of chaos is demonstrated via well-established numerical algorithms that represent the current standard in the field of nonlinear dynamics, i.e., by calculation of the largest Lyapunov exponent and high-resolution 1-D bifurcation diagrams. The achieved numerical results are put into the context of experimental measurement; observed state trajectories prove a one-to-one correspondence between theoretical expectations and practical outputs, i.e., prescribed strange attractors do not represent the chaotic transients. Finally, short term unpredictability of the chaotic flow is demonstrated via calculation of Kaplan-Yorke dimension that is high, i.e., generated waveforms can find interesting applications in the fields of chaotic masking, modulation, or chaos-based cryptography.

Optimizing the Kaplan–Yorke Dimension of Chaotic Oscillators Applying DE and PSO

“…The third chaotic oscillator used in this article is the Lorenz chaotic system [9], which is described by (3), and its design parameters are σ = 0.16, ρ = 45.92, and β = 4. Its numerical simulation provides LE+ = 2.16 and D KY = 2.07, as listed in Table 1, and its phase portrait is shown in Figure 1.…”

“…The initial conditions (x 0 , y 0 , z 0 ) matters to reduce computing time in computing Lyapunov exponents and D KY . In this manner, the appropriate initial conditions are: (ae 0.5 , 0.5, 0.75) for the chaotic oscillator with infinite equilibria (1), (0.5, 0.5, 0.5) for Rössler (2), and (0.1, 0.1, 0.1) for Lorenz (3).…”

“…For instance, the authors in [2] introduced a flexible chaotic system through applying modification in a recent rare chaotic flow which has adjustable D KY , and it is used in a practical application showing a relation between D KY 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 D KY that is high, so that the generated chaotic waveforms can find interesting applications in the fields of chaotic masking, modulation, or chaos-based cryptography.…”

A Modified Grey Wolf Optimizer for Optimum Parameters of Multilayer Type-2 Asymmetric Fuzzy Controller

“…Synchronization chaotic systems have attracted great interest because of their potential applications in various fields such as electronics, signal processing, secure communication, aerospace, and biological systems [70][71][72][73][74][75][76][77]. Synchronization of chaotic systems is the control of the slave system to mimic the behavior of the master system.…”

New Nonlinear Active Element Dedicated to Modeling Chaotic Dynamics with Complex Polynomial Vector Fields

“…Moreover, higher-order polynomial resistors can be used to approximate goniometric functions and connected as the fundamental nonlinearity inside suitable circuit topology to generate the so-called multiscroll [ 6 , 7 , 8 ] or multigrid strange attractors [ 9 , 10 ]. By generalization of such a design approach, very simple chaotic systems with a fully passive ladder filter working in the trans-immittance regime can be constructed [ 11 ]. So far, it seems that this is also the simplest way to practically implement the so-called jerky (motion or Newtonian) dynamics, i.e., autonomous deterministic system defined by a single third-order ordinary differential equation [ 12 , 13 , 14 , 15 ].…”