Related papers: The Gardner problem and cycle slipping bifurcation…
The lock-in frequency and lock-in range concepts were introduced in 1966 by Floyd Gardner to describe the frequency differences of phase-locked loop based circuit for which the loop can acquire lock within one beat, i.e. without cycle…
In the present work, a second-order PLL with lead-lag loop filter and triangular phase detector characteristic is analysed. An exact value of the conservative lock-in range is obtained for the considered model. The solution is based on…
Phase-locked loops (PLL), Costas loops and other synchronizing circuits are featured by the presence of a nonlinear phase detector, described by a periodic nonlinearity. In general, nonlinearities can cause complex behavior of the system…
Charge pump phase-locked loop with phase-frequency detector (CP-PLL) is an electrical circuit, widely used in digital systems for frequency synthesis and synchronization of the clock signals. In this paper a non-linear second-order model of…
Phase-locked solutions of coupled oscillators are studied with asymmetric coupling strengths or inhomogeneous natural frequencies. The solutions show remarkable profiles of phase lags from the pacemaker corresponding to the ratio of upward…
In this paper, we consider an overdetermined problem of Serrin-type for a two-phase elliptic operator with piecewise constant coefficients. We show the existence of infinitely many branches of nontrivial symmetry breaking solutions which…
In the present work PLL-based circuits with sinusoidal phase detector characteristic and active proportionally-integrating (PI) filter are considered. The notion of lock-in range -- an important characteristic of PLL-based circuits, which…
In this paper a fairly complete mathematical model of CP-PLL, which reliable enough to serve as a tool for credible analysis of dynamical properties of these circuits, is studied. We refine relevant mathematical definitions of the hold-in…
In a recent work it was suggested that the number of limit cycles in a piecewise-linear system could be closely related to the number of zones, that is the number of parts of the phase plane where the system is linear. In this note we…
Phase-locked loops (PLLs) are now widely used in communication systems and have been a classic system for more than 60 years. Well-known mathematical models of such systems are constructed in a number of approximations, so questions about…
The main purpose of this article is to study from the geometric point of view the problem of limit cycles bifurcation of perturbed completely integrable systems.
The extended definition of the polynomial B-splines may give a chance to improve the results obtained by the classical cubic polynomial B-splines. Determination of the optimum value of the extension parameter can be achieved by scanning…
Exact solutions of the linear water-wave problem describing oblique waves over a submerged horizontal cylinder of small (but otherwise fairly arbitrary) cross-section in a two-layer fluid are constructed in the form of convergent series in…
This paper investigates the multiplicity and the number of limit cycles for planar piecewise linear system divided into two regions by a straight line and each linear subsystem has a node. Through constructing Poincare half maps and a…
We prove well-posedness of solution to the nonlinear Schr\"{o}dinger equation associated to the twisted Laplacian on $\C^n$ for a general class of nonlinearities including power type with subcritical case $0\leq \alpha<\frac{2}{n-1}$, see…
We address a two-dimensional nonlinear elliptic problem with a finite-amplitude periodic potential. For a class of separable symmetric potentials, we study the bifurcation of the first band gap in the spectrum of the linear Schr\"{o}dinger…
The exponential B-spline basis function set is used to develop a collocation method for some initial boundary value problems (IBVPs) to the Gardner equation. The Gardner equation has two nonlinear terms, namely quadratic and cubic ones. The…
The classical Dirichlet problem for a second-order strongly elliptic system with constant coefficients in a Jordan domain is considered. We show that the solution of the problem can be represented as a functional series in powers of the…
In this paper, we study the problem of limit cycle bifurcation in two piecewise polynomial systems of Li\'enard type with multiple parameters. Based on the developed Melnikov function theory, we obtain the maximum number of limit cycles of…
In a phase-locked loop (PLL) synchronized inverter, due to the strong nonlinear coupling between the PLL's parame-ters and the operation power angle, the equivalent damping coefficient will quickly deteriorate while the power angle is close…