Transistor-based chaotic oscillator

• Dr. Ludovico Minati, Tokyo Institute of Technology, Tokyo, Japan

• Prof. Mattia Frasca, DIEEI, University of Catania, Catania, Italy

Figure 1: Representative realization of two coupled single-transistor oscillators, including variable resistors for controlling dynamics and coupling strength, and readout amplifiers, in a hybrid package (Minati et al., 2017).

Transistor-based chaotic oscillators are elementary electronic circuits which can generate strikingly diverse chaotic signals, and whose only active and non-linear components are transistors: usually, but not exclusively, discrete bipolar-junction transistors (BJTs). They can be either non-autonomous, i.e., driven by an external forcing signal, or autonomous, i.e., operating in the absence of external inputs (besides a DC power source).

These circuits are of interest mainly due to the following reasons:

• They tend to be structurally simple, i.e., often have a total number of elements comparable with well-known harmonic and relaxation oscillators, and are therefore easy and convenient to realize and investigate experimentally (Buscarino et al., 2014), as exemplified in Fig. 1;
• Nevertheless, they recapitulate a vast repertoire of dynamical behaviors, including funnel-like, phase-coherent and spiking attractor geometries, and can generate rich bifurcation structures as a function of the control parameters (Ott, 2002; Minati, 2014; Minati et al., 2017);
• They are straightforward to couple, diffusively or unidirectionally, and when partially synchronized they readily give rise to emergent patterns influenced by the presence of component tolerances and non-ideal characteristics (Fortuna and Frasca, 2007; Minati, 2014b);
• They diversely exploit various aspects of transistor physics, including not only the main non-linearities inherent in the junction equations but also more subtle aspects such as the Early effect (Hanias et al., 2010);
• Unlike the "typical" scenario wherein a given system of equations is implemented via operational amplifiers and analog multipliers, they embody a different approach to engineering chaotic oscillators, which, rather than being predicated on a given system form, starts from the physical properties of a basic device, leading to much smaller circuit size.

Here, some examples are presented without any attempt to comprehensively survey this area.

Non-autonomous transistor-based chaotic oscillators

Figure 2: A non-autonomous transistor-based chaotic oscillator (Lindberg et al., 2005). R1=1 k$\Omega$, R2=994 k$\Omega$, C1=4.7 nF, C2=1.1 nF, VS denotes a sinusoidal voltage source having an amplitude 10 V and a frequency of 10 kHz, Q1 type 2N2222A.

Figure 3: A chaotic Colpitts oscillator (Kennedy, 1994). Components: VCC=5 V, RL=35 $\Omega$, L=98.5 $\mu$H, C1=54 nF, C2=54 nF, REE=400 $\Omega$, VEE=-5 V, Q1 type 2N2222A.

Non-autonomous transistor-based chaotic oscillators typically consist of one or two transistors, some passive components (i.e., inductors, capacitors, and resistors) and a signal source (usually sinusoidal). An example is illustrated in Fig. 2. This circuit consists of an ideal sinusoidal oscillator VS, two resistors, two capacitors, and a BJT (Lindberg et al., 2005). The frequency and amplitude of VS both act as control parameters, qualitatively influencing the behavior of the circuit, including the transition between periodic, quasi-periodic and chaotic dynamics. In particular, chaos arises when an antagonistic behavior between two subparts of the circuit ensues. The capacitor C2 is charged through a large resistor R2 by the left subpart of the circuit, following a time constant that is large compared to the period of the sinusoidal oscillator VS. However, when the transistor Q1 is switched on, it effectively short-circuits C2. The charging of C2 is thus disturbed by the oscillation at the collector of Q1 (connected to an RC circuit driven by VS). This results in irregular switching times of the transistor and waveforms of the voltage across C2.

Several other non-autonomous transistor-based chaotic oscillators have been proposed (Hanias and Tombras, 2009; Hanias et al., 2010). They are often referred to as RLT (resistor-inductor-transistor) circuits, and their operation mechanisms resemble those of RLD (resistor-inductor-diode) chaotic circuits (Murali et al., 1994).

Figure 4: A chaotic Hartley oscillator (Kvarda, 2002). Components: V1=3.75 V, R1=1.5 $\Omega$, R2=1.5 $\Omega$, RL=76 $\Omega$, L1=100 $\mu$H, L2=100 $\mu$H, C1=22 nF, RC=95.5 $\Omega$, Q1 type 2N3906.

Autonomous transistor-based chaotic oscillators

Chaos may be produced in transistor-based chaotic oscillators also without the need for an external forcing signal. These circuits thus operate autonomously. In such circuits, chaos is often obtained through a bifurcation from a periodic regime, an observation which paves the way for two different strategies adopted in the design of autonomous transistor-based chaotic oscillators (Ott, 2002). In the first case, they are obtained by operating classical, i.e., periodic, oscillators with different parameters, while in the second case, further circuitry is added to the original structure of a classical oscillator.

A paradigmatic example of the first scenario is the chaotic behavior discovered in the Colpitts oscillator by Kennedy (1994). The circuit diagram is given in Fig. 3. With the parameters shown therein, chaotic behavior is observed. Notably, with minor changes, the circuit may be adapted for ultrahigh-frequency applications, as shown by Li et al. (2013). Another classical oscillator wherein chaotic behavior may be observed for selected values of the components is the Hartley oscillator (Kvarda, 2002; Tchitnga et al., 2012), shown in Fig. 4. In the second approach, a known topology of a periodic oscillator is modified with the inclusion of some additional components to generate chaos. For instance, the chaotic circuit by Elwakil and Kennedy (2000) is obtained by modifying a relaxation oscillator, while the one by Keuninckx et al. (2015) derives from a resistor-capacitor shift oscillator combined with a nonlinear subcircuit including a second transistor.

Rulkov and Volkovskii (2001), instead, adopt a different approach. They consider a pulse oscillator triggered by a chaotic map to generate pulses at chaotic time intervals. The resulting circuit is composed of a blocking oscillator and two additional passive components (a capacitor and an inductor). Yet another circuit worth mentioning is the two-transistor circuit introduced by Matsumoto et al. (1986). This circuit implements the double-scroll Chua's attractor by using a nonlinearity based on two transistors.

Atypical transistor-based chaotic oscillators

Figure 5: Bit-string encoding the connections and component values of a transistor-based circuit, potentially representing a viable chaotic oscillator. The R, C/L and Q regions code for a resistor, multiple capacitors or inductors, and two transistors. VAL denotes value, Nx connection node (Minati et al., 2017).

The notion of “atypical transistor-based chaotic oscillator” was introduced in recognition of the fact that a systematic approach to synthesizing these circuits is presently missing; however, chaos is a relatively common occurrence in viable transistor-based oscillators, due to the inherent non-linearity of the device. To implement a form of “in-silico serendipity", then, an arbitrary circuit can be encoded in a string of 50-100 bits, representing, as shown in Fig. 5, the connections and values of a small number of reactive elements alongside one or two transistors. The resulting search problem can be handled via suitable heuristics, for example in the context of a random search. In order to enhance the probability of observing chaos, a variable resistor (the only resistor found in these circuits) can be connected in series with the DC power source to act as the main control parameter. An "atypical" circuit is thus one discovered using this approach and having such a tunable series resistor. Due to the resulting structural diversity, multiple route-to-chaos mechanisms are observed, including chaotic bifurcations, intermittency and quasiperiodicity, the latter being in particular related to the interplay and "competition" between oscillations at unrelated frequencies, made possible by the presence of multiple LC combinations (Ott, 2002; Minati, 2014; Minati et al., 2017).

In two recent studies conducted using this approach, several tens of these circuits were discovered. At a minimum, two inductors, one capacitor, and one transistor appear necessary to generate chaos, and the topology shown in Fig. 6 (left) is found recurrently. As a function of the component values, it can generate funnel-like, phase-coherent and other attractors, over a frequency range depending on the physical devices (Minati, 2014; Minati et al., 2017). Numerical analysis of this circuit shows that the transistor can be represented simply with a DC voltage source and a non-linear, voltage-dependent current source; however, it is also essential to represent the capacitance at the collector node, yielding a four-dimensional system. By generalizing the circuit to include fractal resonant networks in place of the inductors, high-dimensional dynamics and hyperchaos can be generated (Minati et al., 2018).

On the other hand, more complex circuits can recapitulate certain other cases of particular interest. For example, a two-transistor circuit can produce trains of irregularly-spaced discrete spikes resembling neural activity; structurally, this circuit is considerably more straightforward than the majority of electronic neural models, but attains this at the price of losing the physiological significance of the electrical variables (Fig 6, mid). A further circuit of this kind is among the simplest transistor-based circuits capable of generating the dual-scroll attractor, often associated with implementations of Chua's circuit (Fig 6, right).

Figure 6: Three representative "atypical" chaotic oscillators (Minati et al., 2017) and their dynamics. Components, left: R=700-4500 $\Omega$, C=150-390 pF, L1=15-150 $\mu$H and L2=33-220 $\mu$H; mid: R=210-230 $\Omega$, L1=15 $\mu$H, L2=68 $\mu$H, L3=150 $\mu$H, C=470 pF, right: R=940-1020 $\Omega$, L1=22 $\mu$H, L2=220 $\mu$H, C=470 pF. For all, VCC=5 V and Qx type PRF949.

References

• A. Buscarino, L. Fortuna, M. Frasca, and G. Sciuto (2014), A Concise Guide to Chaotic Electronic Circuits, Springer.
• A.S. Elwakil, and M.P. Kennedy (2000), A low-voltage, low-power, chaotic oscillator, derived from a relaxation oscillator, Microelectronics J 31, 459-468.
• L. Fortuna, and M. Frasca (2007), Experimental synchronization of single-transistor-based chaotic circuits, Chaos 17, 043118.
• M.P. Hanias, I.L. Giannis, and G.S. Tombras (2010), Chaotic operation by a single transistor circuit in the reverse active region, Chaos 20, 013105.
• M.P. Hanias, and G.S. Tombras (2009), Time series analysis in a single transistor chaotic circuit, Chaos Solitons Fractals 40, 246-256.
• M.P. Kennedy (1994), Chaos in the Colpitts oscillator, IEEE Trans. Circuits Syst. I 41, 771-774.
• L. Keuninckx, G. Van der Sande, and J. Danckaert (2015), Simple Two-Transistor Single-Supply Resistor–Capacitor Chaotic Oscillator, IEEE Trans. Circuits Syst. II 62, 891-895.
• P. Kvarda (2002), Chaos in Hartley's oscillator, Int. J Bifurcat. Chaos 12, 2229-2232.
• J.X. Li, Y.C. Wang, and C. Ma (2013), Experimental demonstration of 1.5 GHz chaos generation using an improved Colpitts oscillator, Nonlinear Dyn. 72, 575–580.
• E. Lindberg, K. Murali, and A. Tamasevicius (2005), The smallest transistor-based nonautonomous chaotic circuit, IEEE Trans. Circuits Syst. II 52, 661-664.
• T. Matsumoto, L. Chua, and K. Tokumasu (1986), Double scroll via a two-transistor circuit, IEEE Trans. Circuits Syst. 33, 828-835.
• L. Minati (2014), Experimental dynamical characterization of five autonomous chaotic oscillators with tunable series resistance, Chaos 24, 033110.
• L. Minati (2014), Experimental synchronization of chaos in a large ring of mutually coupled single-transistor oscillators: Phase, amplitude, and clustering effects, Chaos 24, 043108.
• L. Minati, M. Frasca, G. Giustolisi, P. Oświȩcimka, and S. Drożdż (2018), High-dimensional dynamics in a single-transistor oscillator containing Feynman-Sierpiński resonators: Effect of fractal depth and irregularity, Chaos 28, 093112.
• L. Minati, M. Frasca, P. Oświȩcimka, L. Faes, and S. Drożdż (2017), Atypical transistor-based chaotic oscillators: Design, realization, and diversity, Chaos 27, 073113.
• K. Murali, M. Lakshmanan, and L.O. Chua (1994), The simplest dissipative nonautonomous chaotic circuit, IEEE Trans. Circuits Syst. I 41, 462-463.
• E. Ott (2002), Chaos in Dynamical Systems, Cambridge University Press.
• N. Rulkov, and A. Volkovskii (2001), Generation of broad-band chaos using blocking oscillator, IEEE Trans. Circuits Syst. I 48, 673-679.
• R. Tchitnga, H.B. Fotsin, B. Nana, P.H. Louodop Fotso, and P. Woafo (2012), Hartley’s oscillator: The simplest chaotic two-component circuit, Chaos Solitons Fractals 45, 306-313.

See also

Attractor, Bifurcation, Dynamical systems, Chaos, Communicating with chaos, Controlling chaos, Fractals, Hyperchaos, Neuron, Oscillators, Periodic orbit, Quasiperiodic oscillations, Resonance, Synchronization.

Sponsored by: Alessandro Torcini, Istituto dei Sistemi Complessi - CNR - Firenze, Italy

Reviewed by: Dr. Pietro DeLellis, University of Naples Federico II, Naples, Italy

Reviewed by: Anonymous (via Alessandro Torcini, Istituto dei Sistemi Complessi - CNR - Firenze, Italy)

Accepted on: 2019-05-24 09:06:11 GMT