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Abstract
By using one-dimensional (1-D) map methods, some lossless transmission line circuits with a short at one side terminal have been actively studied. Bifurcation results or chaotic states in the circuits have been reported. On the other hand, many weak or strong definitions such that a 1-D map is mathematically chaotic are still being studied. In such definitions, the definition of formal chaos is well known as being the most traditional and most definite. However, formal chaos existences have not been rigorously proven in such circuits. In this paper, a general lossless transmission circuit is considered first with a dc bias voltage source in series with a load resistor at one side terminal and with a three-segment piecewise linear resistor at another side terminal. Secondly, the method for deriving a 1-D map describing the behavior of the circuit is summarized. Thirdly, to provide a basis of chaotic application for the 1-D map, the mathematical definition of formal chaos and the sufficient conditions of the existence of formal chaos are discussed. Furthermore, by using Maple, formal chaos existences and bifurcation behavior of 1-D maps are presented. By using the Lyapunov exponent, the observability of formal chaos in such bifurcation processes is outlined. Finally, the principal results and the future works are summarized.
Highlights
Transmission line circuits, switched capacitor circuits, neuron model circuits, and constrained circuits are notable as the nonlinear circuits of which behavior may be described by one-dimensional (1-D) maps [1,2,3,4,5,6,7,8,9,10,11]
A 1-D discrete dynamical system or 1-D map φ on M is said to be chaotic if there exists an invariant subset Λ ⊂ M on which some iterates of φ is topologically conjugate to the shift dynamics (σ, Σm ) with m symbols
Instead of H, focusing on H 2 of which the circuit parameters are expected to have the broader range implying that H 2 satisfies Theorem 3, we provide Theorem 5 to guarantee that the dynamics of H 2 on I is topologically conjugate with respect to the shift dynamics (σ, Σ2 ) with two symbols
Summary
Transmission line circuits, switched capacitor circuits, neuron model circuits, and constrained circuits are notable as the nonlinear circuits of which behavior may be described by one-dimensional (1-D) maps [1,2,3,4,5,6,7,8,9,10,11]. In [8,9,10,11], by using a 1-D map, analytical bifurcation results or chaotic states (except for formal chaos) are reported in terms of numerical simulation. Formal chaos and observable chaos existence [13,18,20] have not been rigorously proven in such lossless transmission line circuits in [5,6,7,8,9,10,11]. In. Section 4, in order to provide a basis of chaotic application for the 1-D map (or the transmission circuit), we discuss the mathematical definition of formal chaos and the sufficient conditions of the existence of formal chaos for generating 1-D maps. We mention the possibility of designing imperfect transmission lines with parasitic effects and nonidealities inside real integrated devices generating high frequency ranges and chaotic oscillations without the use of additional capacitors or inductors
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