**The Dirac oscillator**

Dirac’s free particle equation originated in an endeavor to precise linearly the relativistic quadratic relation between energy and momentum. The authors introduce a Dirac equation that, besides the momentum, is additionally linear within the coordinates. They decision it the Dirac generator as a result of within the nonrelativistic limit it becomes a harmonic generator with a really robust spin-orbit coupling term. The eigenstates associated eigenvalues of the Dirac generator is obtained in an elementary fashion, with the degeneracy of the latter being quite totally different from that of the standard generator. They shortly mention the symmetry Lie pure mathematics accountable for this degeneracy and therefore the generalisation of the matter to many-particle systems. **[1]**

**Optoelectronic microwave oscillator**

We describe a completely unique generator that converts continuous lightweight energy into stable and spectrally pure microwave signals. This optoelectronic microwave generator consists of a pump optical device Associate in Nursingd a electric circuit together with an intensity modulator, Associate in Nursing glass fiber electric circuit, a photodetector, Associate in Nursing electronic equipment, and a filter. we tend to develop a quasi-linear theory and acquire expressions for the edge condition, the amplitude, the frequency, the road dimension, and also the spectral power density of the oscillation. we tend to additionally gift experimental knowledge to match with the theoretical results. Our findings indicate that the optoelectronic microwave generator will generate ultrastable, spectrally pure microwave reference signals up to seventy five gigacycle with a section noise below -140 dBc/Hz at ten kilohertz.** [2]**

**Chaos in the Colpitts oscillator**

In this work, we have a tendency to gift experimental results and SPICE simulations of chaos during a Colpitts generator. we have a tendency to show that the nonlinear dynamics of this generator could also be sculpturesque by a third-order autonomous continuous-time circuit consisting of a linear electrical device, 2 linear capacitors, 2 linear resistors, 2 freelance voltage sources, a linear current-controlled current supply, and one voltage-controlled nonlinear electrical device. The nonlinear electrical device contains a two-segment piecewise-linear refugee characteristic. With the suitable selection of parameters, the piecewise-linear circuit model contains a positive Lyapunov exponent. **[3]**

**Light-fuelled freestyle self-oscillators**

Self-oscillation may be a development wherever AN object sustains motion upon non-periodic stimulant. It happens normally in nature, a number of examples being heartbeat, ocean waves and undulation of leaves. Stimuli-responsive materials enable making artificial self-oscillators fuelled by completely different varieties of energy, e.g. heat, lightweight and chemicals, showing nice potential for applications in power generation, autonomous mass transport, and self-propelled micro-robotics. However, most of the self-oscillators area unit supported bending deformation, thereby limiting their potentialities of being enforced in sensible applications. **[4]**

**Analyzing of Nonlinear Generalized Duffing Oscillators Using the Equivalent Linearization Method with a Weighted Averaging**

The generalized Duffing generator is investigated during this paper by exploitation the Equivalent Linearization methodology with a weighted averaging. Applying of the Equivalent Linearization methodology during which the averaging price is calculated during a new means known as the weighted averaging price by introducing a weighted constant perform, the amplitude-frequency relationship of the generator is obtained during a closed-form. The obtained solutions are compared with approximate analytical solutions, precise solutions and numerical solutions. Comparisons show the dependability of this solutions.** [5]**

**Reference**

**[1]** Moshinsky, M. and Szczepaniak, A., 1989. The dirac oscillator. Journal of Physics A: Mathematical and General, 22(17), (Web Link)

**[2]** Yao, X.S. and Maleki, L., 1996. Optoelectronic microwave oscillator. JOSA B, 13(8), (Web Link)

**[3]** Kennedy, M.P., 1994. Chaos in the Colpitts oscillator. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 41(11), (Web Link)

**[4]** Light-fuelled freestyle self-oscillators

Hao Zeng, Markus Lahikainen, Li Liu, Zafar Ahmed, Owies M. Wani, Meng Wang, Hong Yang & Arri Priimagi

Nature Communications volume 10, (Web Link)

**[5]** Hieu, D. V. and Hai, N. Q. (2018) “Analyzing of Nonlinear Generalized Duffing Oscillators Using the Equivalent Linearization Method with a Weighted Averaging”, Asian Research Journal of Mathematics, 9(1) (Web Link)