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Summary: In the paper, a nonlinear analysis of RC phase‐shift oscillator with operational amplifier is done. Using Kirchhoff's laws, a nonlinear system of three differential equations that describes the behavior of the oscillator is obtained. This system is analyzed using the Poincaré–Andronov–Hopf bifurcation theory. The basic principles of the Poincaré–Andronov–Hopf bifurcation theory are presented in advance, including the Center Manifold Theory and the Theory of Normal Forms. It was found that, under certain conditions, the oscillator system has a limit cycle, which means that the oscillator under consideration generates periodic oscillations. The period, amplitude, and stability of the generated periodic oscillations are determined theoretically. The obtained analytical results are confirmed by a numerical analysis of the nonlinear system of differential equations performed by MATLAB. [ABSTRACT FROM AUTHOR]

In this paper, we present an improved method for analyzing the stability of the nonlinear hybrid dynamical systems (NHDSs) using a monodromy matrix. We define an n‐dimensional NHDS and focus on an orbit whose initial values exist near the periodic orbit. Because the system has nonlinearity, the circuit equation cannot be expressed in the form of a linear ordinary differential equation, meaning that the existing monodromy‐matrix‐based stability analysis method cannot be used because it requires a matrix exponential. Therefore, we propose a general theory for calculating orbital perturbations during Poincaré observation. Perturbation through switching events is expressed by a state‐transition matrix, which we refer to as the saltation matrix. By computing the perturbations, we obtain the monodromy matrix, whose characteristic multipliers denote the stability of the periodic orbit. We apply the proposed algorithm to an interrupted electric circuit with a nonlinear characteristic to confirm its validity. © 2018 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc. [ABSTRACT FROM AUTHOR]

Abstract: We consider several different bidirectional Whitham equations that have recently appeared in the literature. Each of these models combines the full two‐way dispersion relation from the incompressible Euler equations with a canonical shallow water nonlinearity, providing nonlocal model equations that may be expected to exhibit some of the interesting high‐frequency phenomena present in the Euler equations that standard “long‐wave” theories fail to capture. Of particular interest here is the existence and stability of periodic traveling wave solutions in such models. Using numerical bifurcation techniques, we construct global bifurcation diagrams for each system and compare the global structure of branches, together with the possibility of bifurcation branches terminating in a “highest” singular (peaked/cusped) wave. We also numerically approximate the stability spectrum along these bifurcation branches and compare the stability predictions of these models. Our results confirm a number of analytical results concerning the stability of asymptotically small waves in these models and provide new insights into the existence and stability of large amplitude waves. [ABSTRACT FROM AUTHOR]

We present a nonstandard discretization method related to the methods of Mickens and Elaydi and Sacker for converting single species population models from continuous time to discrete time. The method falls in the category of the so‐called nonstandard discretization schemes, that is more advantageous than the classical discretization methods (such as adaptive step‐size) since it allows large step sizes. For instance, a large step size could better represent a generation time or a time interval between empirical measurements. Examples of single‐species models with and without negative density dependence, with an Allee effect, and with an alternative positive stable equilibrium (predator pit) are studied. Comparative analyses of bifurcations of ordinary differential equations and difference equations show how the new discretization proposed here preserves the dynamical properties of the continuous‐time models. Recommendations for Resource Managers: The discretization method we propose preserves the original dynamic properties of the continuous model, in the sense of equilibria, their stability, and bifurcation characteristics. Unlike the traditional numerical methods that are widely used in ecology, the dynamical consistency of our method does not depend on the size of the step size used.The discretization method we propose produces solution trajectories in a remarkable agreement with those of the corresponding continuous models irrespective of the size of the time interval used.Results presented here will be important to future ecological studies that seek to evaluate the pervasiveness and strength of negative density dependence as well as Allee effects, along with the prospects of alternative stable states, in natural populations. [ABSTRACT FROM AUTHOR]