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In this paper, we investigate numerically the dynamic snap-through motion of a shallow arch micro-beam that is electrostatically excited through three separated electrodes. It is excited by a dc voltage and a time-varying signal with a frequency tuned near its first in-plane frequency in both initial curvature and initial-counter curvature configurations. A reduced-order model based on the Galerkin method is utilized to discretize the governing equation of motion and to analyze the static and dynamic responses of the micro-beam. The phase portraits obtained from simulations reveal the presence of a stable response that oscillates back and forth around single and double equilibria until it eventually settles in a steady state zone near the first equilibrium point or encloses both equilibria. This can be tracked using a Long-Time Integration (LTI) and recording the time–history signal for a long period. Remarkably, this behavior can be controlled by adjusting the initial operating conditions mainly the signal frequency. We also observe the emergence of a chaotic attractor in the vicinity of the resonance frequency when the shallow arch beam is excited while it is in its initial-counter curvature. The simulated FFTs indicate the presence of the chaotic attractor in the vicinity of primary resonance due to dynamic snap-through motion or large excitation between two equilibria. We observed the existence of periodic orbits with single loop (P-1) and an full chaotic behavior around the two equilibria. The existence of this phenomenon can be attributed to the fact that the excitation voltage is positioned close to the snap-through threshold. [ABSTRACT FROM AUTHOR]

This paper provides a novel approach to studying the chaos control and synchronization of fractional-order hybrid Darcy–Brinkman systems (FOHDBs). We use the truncated Galerkin method to transform the system of partial differential equations (PDEs) into ordinary differential equations (ODEs) based on Fourier modes in a nonlinear low-dimensional Brinkman model. In the nonlinear complex system, hybrid nanoparticles and fluid flow physical characteristics are considered to be external disturbances. As a result, the implications of model uncertainty and external disruptions are fully absorbed into the problem, further complicating it and resulting in highly complex and unpredictable behaviors. In addition, To assure the occurrence of the sliding motion in a finite amount of time, an appropriate robust fractional sliding mode technique is proposed. Sliding motion is shown to occur in finite time using the fractional Lyapunov stability. Following that, we discussed control techniques for achieving infinitesimally close to an equilibrium of chaotic states and tracking errors. It is important to note that FOHDBs and external stochastic disturbances can be controlled using the proposed fractional nonsingular terminal sliding mode control technique. Finally, the effectiveness and applicability of the suggested finite-time control technique are graphically illustrated by the analytical and numerical simulations. [ABSTRACT FROM AUTHOR]