Discretization analysis of bifurcation based nonlinear amplifiers

Recently, for modeling biological amplification processes, nonlinear amplifiers based on the supercritical Andronov-Hopf bifurcation have been widely analyzed analytically. For technical realizations, digital systems have become the most relevant systems in signal processing applications. The underlying continuous-time systems are transferred to the discrete-time domain using numerical integration methods. Within this contribution, effects on the qualitative behavior of the Andronov-Hopf bifurcation based systems concerning numerical integration methods are analyzed. It is shown exemplarily that explicit Runge-Kutta methods transform the truncated normalform equation of the Andronov-Hopf bifurcation into the normalform equation of the Neimark-Sacker bifurcation. Dependent on the order of the integration method, higher order terms are added during this transformation. A rescaled normalform equation of the Neimark-Sacker bifurcation is introduced that allows a parametric design of a discrete-time system which corresponds to the rescaled Andronov-Hopf system. This system approximates the characteristics of the rescaled Hopf-type amplifier for a large range of parameters. The natural frequency and the peak amplitude are preserved for every set of parameters. The Neimark-Sacker bifurcation based systems avoid large computational effort that would be caused by applying higher order integration methods to the continuous-time normalform equations. © Author(s) 2017

Numerical analysis of dynamical systems with codimension two singularities

Páez Chávez JN. Numerical analysis of dynamical systems with codimension two singularities. Bielefeld (Germany): Bielefeld University; 2009

Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. Part 1: The ODE connection and its implications for algorithm development in computational fluid dynamics

Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit

Dynamic Investigation of the Hunting Motion of a Railway Bogie in a Curved Track via Bifurcation Analysis

The main purpose of this paper is to analyze and compare the Hopf bifurcation behavior of a two-axle railway bogie and a dual wheelset in the presence of nonlinearities, which are yaw damping forces in the longitudinal suspension system and heuristic creep model of the wheel-rail contact including dead-zone clearance, while running on a curved track. Two-axle railway bogie and dual wheelset were modeled using 12-DOF and 8-DOF system with considering lateral, vertical, roll, and yaw motions. By utilizing Lyapunov’s indirect method, the critical hunting speeds related to these models are evaluated as track radius changes. Hunting defined as the lateral vibration of the wheelset with a large domain was characterized by a limit cycle-type oscillation behavior. Influence of the curved track radius on the lateral displacement of the leading wheelset was also investigated through 2D bifurcation diagram, which is employed in the design of a stable model. Frequency power spectra at critical speeds, which are related to the subcritical and supercritical bifurcations, were represented by comparing the two-axle bogie and dual wheelset model. The evaluated accuracy to predict the critical hunting speed is higher and the hunting frequency in unstable region is lower compared to the dual wheelset model