The amplitude system for a Simultaneous short-wave Turing and long-wave Hopf instability

<p style='text-indent:20px;'>We consider reaction-diffusion systems for which the trivial solution simultaneously becomes unstable via a short-wave Turing and a long-wave Hopf instability. The Brusseletor, Gierer-Meinhardt system and Schnakenberg model are prototype biological pattern forming systems which show this kind of behavior for certain parameter regimes. In this paper we prove the validity of the amplitude system associated to this kind of instability. Our analytical approach is based on the use of mode filters and normal form transformations. The amplitude system allows us an efficient numerical simulation of the original multiple scaling problems close to the instability.</p>

Design, simulation, and adjustment of the longitudinal–torsional composite integrated ultrasonic variable amplitude system

For the longitudinal–torsional composite integrated ultrasonic variable amplitude system with the longitudinal–torsional resonance produced by helical slots in the cone of the launch section, the theoretical design, finite element simulation, and experimental study are carried out. The performed analysis of the helical slot structural characteristics revealed that slot depth, helical angle, and slot width have a strong effect on the natural frequency and ratio of the torsional–longitudinal amplitude. A credible technique for adjusting the deviation of the actual system natural frequency from the theoretical design value by traditional method is proposed based on the adjustment theory of non-uniform medium adopted. The experimentally proved reduction of the above deviation strongly suggests that the approach proposed in this article provides effective theoretical guidance and experimental basis for the study of longitudinal–torsional composite integral ultrasonic variable amplitude system.

WKB analysis of generalized derivative nonlinear Schrödinger equations without hyperbolicity

We consider the semi-classical limit of nonlinear Schrödinger equations in the presence of both a polynomial nonlinearity and the derivative in space of a polynomial nonlinearity. By working in a class of analytic initial data, we do not have to assume any hyperbolic structure on the (limiting) phase/amplitude system. The solution, its approximation, and the error estimates are considered in time-dependent analytic regularity.

Monitoring Mechanical Vibration Amplitude System Design Based on the PVDF Piezoelectric Film Sensor

Vibration monitoring of machinery and equipment plays a very important role in judging the equipment maintenance and operational status. This paper discussed the mechanical vibration amplitude measurement system scheme based on PVDF piezoelectric sensors. The entire test system is mainly composed of PVDF piezoelectric film sensors, charge amplifiers, data acquisition and processing and display of valid values. It can display the amplitude of vibration signals in real time and alarm. The measurement system has the advantages of a simple structure and convenient operation.

THE FOLD-FLIP BIFURCATION

The fold-flip bifurcation occurs if a map has a fixed point with multipliers +1 and -1 simultaneously. In this paper the normal form of this singularity is calculated explicitly. Both local and global bifurcations of the unfolding are analyzed by exploring a close relationship between the derived normal form and the truncated amplitude system for the fold-Hopf bifurcation of ODEs. Two examples are presented, the generalized Hénon map and an extension of the Lorenz-84 model. In the latter example the first-, second- and third-order derivatives of the Poincaré map are computed using variational equations to find the normal form coefficients.

Experimental Response of a Preloaded Vibro-Impacting Hertzian Contact

This paper concerns the non-linear dynamic response of a vibro-impacting Hertzian contact. Sinusoidal and random external normal forces are considered. We focus on the primary resonance and include vibro-impact responses in order to analyze the main characteristics of the system associated to both Hertzian and contact loss non-linearities. Under very small input amplitude, contact exhibits an almost linear resonance. Linearized resonance frequency and damping ratio are identified. Increasing the external input amplitude, the softening behaviour induced by Hertzian nonlinear stiffness is clearly demonstrated for both sinusoidal and random inputs. For higher input amplitude, system exhibits vibro-impacts. The contact loss non-linearity strongly governs the dynamic behaviour of the system.