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The Van der Pol equation is a paradigmatic model of relaxation oscillations. This remarkable nonlinear phenomenon of self-sustained oscillatory motion underlies important rhythmic processes in nature and electrical engineering. Relaxation…

The effects of nonlinear oscillations in compact stars are attracting considerable current interest. In order to study such phenomena in the framework of fully nonlinear general relativity, highly accurate numerical studies are required. We…

A numerical study is presented of one-dimensional and two-dimensional random lasers as a function of the pumping rate above the threshold for lasing. Depending on the leakiness of the cavity modes, we observe that the stationary lasing…

Pulsars are potentially the most remarkable physical laboratories we will ever use. Although in many senses they are extremely clean systems there are a large number of instabilities and variabilities seen in the emission and rotation of…

Nonlinear damping, the change in damping rate with the amplitude of oscillations plays an important role in many electrical, mechanical and even biological oscillators. In novel technologies such as carbon nanotubes, graphene membranes or…

Van der Pol oscillators are prototypical self-sustaining oscillators which have been used to model nonlinear processes in biological and other classical processes. In this work, we investigate how quantum fluctuations affect phase-locking…

Autonomous oscillators, such as clocks and lasers, produce periodic signals \emph{without} any external frequency reference. In order to sustain stable periodic motions, there needs to be external energy supply as well as nonlinearity built…

We investigate the nonlinear interaction between a relativistically strong laser beam and a plasma in the quantum regime. The collective behavior of the electrons is modeled by a Klein-Gordon equation, which is nonlinearly coupled with the…

Non-linear parametric resonances occur frequently in nature. Here we summarize how they can be studied by means of perturbative methods. We show in particular how resonances can affect the motion of a test particle orbiting in the vicinity…

We study oscillations in quantum cascade lasers due to traveling electric field domains, which are observed both in simulations and experiments. These oscillations occur in a range of negative differential resistance and we clarify the…

Frequency entrainment of continuous-variable oscillators has to date been restrained to the weakly nonlinear regime. Here we overcome this bottleneck and extend frequency entrainment of quantum continuous-variable oscillators to arbitrary…

We discuss negative damping regimes in quantum nano-electromechanical systems formed by coupling a mechanical oscillator to a single-electron transistor (normal or superconducting). Using an analogy to a laser with a tunable atom-field…

A peculiar radiation arising as a result of radiation interference of nonlinear oscillators excited by a monochromatic plane wave field of the incident particle is described. The radiation properties are determined by the fact that a phase…

Beam instabilities and resonances affect the transverse dynamics in particle accelerators and, when encountered, can trigger emittance growth and beam loss. Resonance lines originate from non linear elements and effects in the lattice,…

Optomechanical systems, due to its inherent nonlinear optomechanical coupling, owns rich nonlinear dynamics of different types of motion. The interesting question is that whether there exist some common quantum features to infer the…

For noisy self-sustained oscillators, both reliability, stability of a response to a noisy driving, and coherence understood in the sense of constancy of oscillation frequency belong to the main characteristics. Though the both…

Self-oscillating systems, described in classical dynamics as limit cycles, are emerging as canonical models for driven dissipative nonequilibrium open quantum systems, and as key elements in quantum technology. We consider a family of…

Stochastically driven nonlinear processes limit the number of amplified modes in a natural system due to competitive mode interaction, which is accompanied by loss of coherence when increasing the complexity of system. Specifically, we find…

The analysis of a birhythmic modified van der Pol type oscillator driven by periodic excitation and L\`evy noise shows the possible occurrence of coherence resonance and stochastic resonance. The frequency of the harmonic excitation in the…

We review our recent work leading to steady-state solutions of the semiclassical (Maxwell-Bloch) equations of a laser. These are coupled non-linear partial differential equations in space and time which have previously been solved either by…