A systematic procedure for synthesizing two‐terminal devices with polynomial non‐linearity

Polák

2019

IEEE Access

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.

Section: Description Of Sub-circuitsmentioning

confidence: 99%

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

Polák

2019

IEEE Access

“…Of course, it is also possible to build both analog networks provided in Figure 1 directly. Instead of nonlinear two-ports we must construct a couple of resistors with polynomial AVC; systematic design towards these network elements can be found in [39,40]. Circuitry realization of original MVMS with state vector x = (v 1 , v 2 , i) T is demonstrated by means of Figure 10; i.e., the state variables are voltages across grounded capacitors and current flowing through the inductor.…”

Section: Circuitry Realization Of Mvms-based Chaotic Oscillatorsmentioning

confidence: 99%

Strange Attractors Generated by Multiple-Valued Static Memory Cell with Polynomial Approximation of Resonant Tunneling Diodes

2018

Entropy

This paper brings analysis of the multiple-valued memory system (MVMS) composed by a pair of the resonant tunneling diodes (RTD). Ampere-voltage characteristic (AVC) of both diodes is approximated in operational voltage range as common in practice: by polynomial scalar function. Mathematical model of MVMS represents autonomous deterministic dynamical system with three degrees of freedom and smooth vector field. Based on the very recent results achieved for piecewise-linear MVMS numerical values of the parameters are calculated such that funnel and double spiral chaotic attractor is observed. Existence of such types of strange attractors is proved both numerically by using concept of the largest Lyapunov exponents (LLE) and experimentally by computer-aided simulation of designed lumped circuit using only commercially available active elements.

“…This resistor can be piecewise linear; for example, where a third-order admittance network contains both capacitors and inductors as functional accumulation elements is provided in [ 1 ], while an RC passive ladder network connected as a load for active nonlinear resistors is the primary subject of [ 2 ]. Polynomial resistors with arbitrary degrees [ 3 ] can be connected in parallel with a third-order fully passive admittance network to obtain a robust chaotic oscillator [ 4 , 5 ] as well. 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 ].…”

Section: Introductionmentioning

confidence: 99%

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

Šotner

2019

Entropy

This paper describes evolution of new active element that is able to significantly simplify the design process of lumped chaotic oscillator, especially if the concept of analog computer or state space description is adopted. The major advantage of the proposed active device lies in the incorporation of two fundamental mathematical operations into a single five-port voltage-input current-output element: namely, differentiation and multiplication. The developed active device is verified inside three different synthesis scenarios: circuitry realization of a third-order cyclically symmetrical vector field, hyperchaotic system based on the Lorenz equations and fourth- and fifth-order hyperjerk function. Mentioned cases represent complicated vector fields that cannot be implemented without the necessity of utilizing many active elements. The captured oscilloscope screenshots are compared with numerically integrated trajectories to demonstrate good agreement between theory and measurement.