Abstract
A simple chaotic snap circuit based on a single transistor is presented with tunable damping. It suggests, at present, the simplest chaotic snap circuit in the sense that it requires only 9 devices, which offer the minimum number of devices for a chaotic snap circuit. It also suggests the first and simplest circuit realization of either a four-dimensional (4D) chaotic system or a 4th-order (snap) chaotic system that demonstrates a maximized attractor dimension (D L ) of a parameter set, or of the entire parameter space of the system, at minimized damping. The tendency of an increase in D L until its peak is illustrated by a decrease in damping. It offers the highest attractor dimension in a category of unit-damping snap chaos. As an initial report, a Clapp oscillator is able to exhibit 4D chaos but does not allow snap chaos. The proposed snap circuit embeds two simple mechanisms: (i) a Clapp oscillator as a simple core engine of oscillations avoiding a need for op-amps, and (ii) a single resistor as a remarkably simple realization of adjustable damping for snap chaos. A current-tunable equilibrium exhibits one of the 4 different types, two of which are of an (unstable) spiral saddle equilibrium, whereas the others are of a spiral stable equilibrium. They reveal the first report on either saddle-equilibrium or stable-equilibrium snap chaos based on a single transistor. Multistability and hidden attractors are demonstrated. The simple circuit offers a novel damping-tunable single-transistor-based approach to such rich dynamics of snap flows through various types of self-excited and hidden attractors.
Highlights
In low dimensional systems, three successive time derivatives of a variable (x) are acceleration (x), and jkenrokw(n.x..)as[1(]p,h[a2s]e.space) velocity (x), In high dimensional systems, name, is oafte4nthc-aolrleddersntaimp e(.x.d..)er[i2v]a,twivhe,ertehaosuagthimneodestraivnadtaivrde higher than the third is referred to as hyperjerk [3]
Other hand, it naturally leads to six motivating questions of whether a simple snap circuit based on a single transistor will be adequately realizable for (i) chaos, with: (ii) a maximized attractor dimension (DL) of a parameter set, (iii) a higher value of a maximized DL than that of the existing snap chaos which may be compared at the similar damping, (iv) an initial report of 4D chaos in a Clapp oscillator, (v) an exploitation of a Clapp oscillator with simplicity for a chaotic snap circuit, and (vi) a first stableequilibrium system based on a single transistor
(iv) this paper initially reveals that a Clapp oscillator can demonstrate 4D chaos, but cannot exhibit snap chaos, nor does it allow a snap ordinary differential equations (ODEs) despite its four energy-storage elements that could have enabled a simple snap circuit
Summary
Three successive time derivatives of a variable (x) are acceleration (x), and jkenrokw(n.x..)as[1(]p,h[a2s]e. Other hand, it naturally leads to six motivating questions of whether a simple snap circuit based on a single transistor will be adequately realizable for (i) chaos, with: (ii) a maximized attractor dimension (DL) of a parameter set, (iii) a higher value of a maximized DL than that of the existing snap chaos which may be compared at the (relatively) similar damping, (iv) an initial report of 4D (no-snap) chaos in a Clapp oscillator, (v) an exploitation of a Clapp oscillator with simplicity for a chaotic snap circuit, and (vi) a first stableequilibrium system based on a single transistor. The second mechanism embeds a single resistor RC as a remarkably simple realization of tunable damping for, as will be shown, possible snap chaos, a maximized attractor dimension of a given parameter set, multistability, and rich snap flows through various types of self-excited and hidden attractors. The set of four coupled 1st-order ODEs in (1) can be transferred to a 4D normalized, dimensionless, dynamical model as
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