Abstract
This paper presents a simple oscillator using a battery and a second order memristor without the energy storage elements inductor and capacitor. The oscillating mechanism of the proposed circuit has been explained via Hopf bifurcation theorem, small signal model, local activity principle and edge of chaos theorem. This paper can be also used as a reference for explaining the intimate relationship between the super-critical Hopf bifurcation phenomenon and the edge of chaos.
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- 1.
The circuit diagram of the Wien-bridge oscillator can be found from the following link http://www.circuitstoday.com/wien-bridge-oscillator.
- 2.
The DC V-I curve in Fig. 5d for negative voltage \( (V \le 0) \) is just the reflected (odd-symmetric) mirror image about the origin V = 0 over the positive input voltage \( (V \ge 0) \) region.
- 3.
We would like to caution the readers that the DC current I ext is the input in Chua et al. (2012a, b), and the two small-signal equivalent circuits of the potassium ion-channel memristor and the sodium ion-channel memristor in the HH model are connected parallel. Hence, the Eigen values of the Jacobian matrix are identical to the poles of the small-signal impedance \( Z(s,I) \triangleq \frac{V(s)}{I(s)} = \frac{1}{Y(s)} \), or equivalently, the zeros of the admittance Y(s). In the 2nd-order memristor case, the input is a DC voltage V and the two small-signal circuit components shown in Fig. 8 are connected in series. It follows that the poles of the admittance function \( Y(s,V) \triangleq \frac{I(s)}{V(s)} \) of the second-order memristor in Fig. 1c are equivalent to the Eigen values of the Jacobian matrix (47).
- 4.
In contrast, the second-order sodium ion-channel memristor oscillator circuit presented in Fig. 40 of Chua (2014) spawns an unstable sinusoidal oscillation near a boundary of the edge of chaos of the sodium memristor via a sub-critical Hopf bifurcation, a much more subtle bifurcation phenomenon.
References
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Acknowledgements
The authors would like to acknowledge financial support from the USA Air force office of Scientific Research under Grant number FA9550-13-1-0136 and from the European Commission Marie Curie Fellowship and two National Research Foundation of Korea (NRF) grants funded by the Korea government (2013R1A2A2A01068683 and 2012R1A1A2044078).
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Sah, M.P., Rajamani, V., Mannan, Z.I., Eroglu, A., Kim, H., Chua, L. (2017). A Simple Oscillator Using Memristor. In: Vaidyanathan, S., Volos, C. (eds) Advances in Memristors, Memristive Devices and Systems. Studies in Computational Intelligence, vol 701. Springer, Cham. https://doi.org/10.1007/978-3-319-51724-7_2
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