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The nonlinear capacitor that obeys of a cubic polynomial voltage–charge relation (usually a power series in charge) is introduced. The quantum theory for a mesoscopic electric circuit with charge discreteness is investigated, and the Hamiltonian of a quantum mesoscopic electrical circuit comprised by a linear inductor, a linear resistor and a nonlinear capacitor under the influence of a time-dependent external source is expressed. Using the numerical solution approaches, a good analytic approximate solution for the quantum cubic Duffing equation is found. Based on this, the persistent current is obtained antically. The energy spectrum of such nonlinear electrical circuit has been found. The dependency of the persistent current and spectral property equations to linear and nonlinear parameters is discussed by the numerical simulations method, and the quantum dynamical behavior of these parameters is studied.

Iacopini and Zavattini [Vacuum polarization effects in the ${({\mu }^{-4}\mathrm{\text{He}})}^{+}$ atom and the Born–Infeld electromagnetic theory, Nuovo Cimento B78 (1983) 38–52] proposed a $(p,\tau )$-two-parameter modification of Born–Infeld electrodynamics, in which the classical self-energy for an electron takes a finite value for $p<1$. In this paper, we want to study a cylindrical capacitor from the viewpoint of Iacopini–Zavattini nonlinear electrodynamics analytically. The capacitance, the electrostatic potential energy, and the potential difference between the plates of a cylindrical capacitor are calculated in the framework of Iacopini–Zavattini electrodynamics for two specific values of $p=\frac{1}{2}$ and $p=\frac{3}{4}$. The study of the behavior of a nonlinear cylindrical capacitor in the weak electric fields shows that our results are compatible with the correspondence principle, i.e. we recover the results of Maxwell electrodynamics in the weak field regime. Finally, the invariance of Iacopini–Zavattini nonlinear electrodynamics under the duality transformation is investigated.

We briefly analyze and demonstrate several nonlinear signal generators, according to the principles proposed in [1]. These systems apparently should not oscillate, because they include only capacitors in the positive feedback loops. However, the design attempted to make use of the well-known parasitic elements of the capacitors to create a selective feedback loop with physical capacitors only. The use of the resonant (piezoelectric) load adds a new potential mode of oscillation. The design aimed to make achievable as many as possible frequency points where oscillation modes ay exist, against the rather common belief that a single oscillation mode can be supported at one time in electronic oscillators. For this purpose, variable, independent gains have been provided to the positive feedback loops that correspond to the various modes in the circuit. We describe the schemes, briefly analyze their behavior, show simulation and experimental results, and discuss potential uses. The paper is largely based on [1].