Quantifying the parameter dependent basin of the unsafe regime of asymmetric Lévy-noise-induced critical transitions

In real systems, the unpredictable jump changes of the random environment can induce the critical transitions (CTs) between two non-adjacent states, which are more catastrophic. Taking an asymmetric Lévy-noise-induced tri-stable model with desirable, sub-desirable, and undesirable states as a prototype class of real systems, a prediction of the noise-induced CTs from the desirable state directly to the undesirable one is carried out. We first calculate the region that the current state of the given model is absorbed into the undesirable state based on the escape probability, which is named as the absorbed region. Then, a new concept of the parameter dependent basin of the unsafe regime (PDBUR) under the asymmetric Lévy noise is introduced. It is an efficient tool for approximately quantifying the ranges of the parameters, where the noise-induced CTs from the desirable state directly to the undesirable one may occur. More importantly, it may provide theoretical guidance for us to adopt some measures to avert a noise-induced catastrophic CT. © 2021, The Author(s)

Bifurcation Analysis in a Delay Differential Equations, which Confers a Strong Allee Effect in Escherichia Coli

The paper addresses the bifurcations for a delay differential model with parameters which confers a strong Allee effect in Escherichia coli. Stability and local Hopf bifurcations are analyzed when the delay τ or σ as parameter. It is also found that there is a non-resonant double Hopf bifurcation occur due to the vanishing of the real parts of two pairs of characteristic roots. We transform the original system into a finite dimensional system by the center manifold theory and simplify the system further by the normal form method. Then, we obtain a complete bifurcation diagram of the system. Finally, we provide numerical results to illustrate our conclusions. There are many interesting phenomena, such as attractive quasi-periodic solution and three-dimensional invariant torus

<Contributed Talk 22>Non-hyperbolic Equilibria of SD Oscillator

[Date] November 28 (Mon) - December 2 (Fri), 2011: [Place] Kyoto University Clock Tower Centennial Hall, Kyoto, JAPA

Periodic Solution of a Non-Smooth Double Pendulum with Unilateral Rigid Constrain

In this paper, a double pendulum model is presented with unilateral rigid constraint under harmonic excitation, which leads to be an asymmetric and non-smooth system. By introducing impact recovery matrix, modal analysis, and matrix theory, the analytical expressions of the periodic solutions for unilateral double-collision will be discussed in high-dimensional non-smooth asymmetric system. Firstly, the impact laws are classified in order to detect the existence of periodic solutions of the system. The impact recovery matrix is introduced to transform the impact laws of high-dimensional system into matrix. Furthermore, by use of modal analysis and matrix theory, an invertible transformation is constructed to obtain the parameter conditions for the existence of the impact periodic solution, which simplifies the calculation and can be easily extended to high-dimensional non-smooth system. Hence, the range of physical parameters and the restitution coefficients is calculated theoretically and non-smooth analytic expression of the periodic solution is given, which provides ideas for the study of approximate analytical solutions of high-dimensional non-smooth system. Finally, numerical simulation is carried out to obtain the impact periodic solution of the system with small angle motion

Vibration reduction in beam bridge under moving loads using nonlinear smooth and discontinuous oscillator

The coupled system of smooth and discontinuous absorber and beam bridge under moving loads is constructed in order to detect the effectiveness of smooth and discontinuous absorber. It is worth pointing out that the coupled system contains an irrational restoring force which is a barrier for conventional nonlinear techniques. Hence, the harmonic balance method and Fourier expansion are used to obtain the approximate solutions of the system. The first and the second kind of generalized complete elliptic integrals are introduced. Furthermore, using power flow approach, the performance of smooth and discontinuous absorber in vibration reduction is estimated through the input energy, the dissipated energy, and the damping efficiency. It is interesting that only depending on the value of the smoothness parameter, the efficiency parameter of vibration reduction is optimized. Therefore, smooth and discontinuous absorber can adapt itself to effectively reducing the amplitude of the vibration of the beam bridge, which provides an insight to the understanding of the applications of smooth and discontinuous oscillator in engineering and power flow characteristics in nonlinear system

A Novel Dynamic Absorber with Variable Frequency and Damping

Smoothness and discontinuous (SD) oscillator is a nonlinear oscillator with the variable frequency, whose frequency can be varied with the smoothing parameter. However, how to adjust the smoothing parameter has not been solved in the actual device. In this paper, the shape memory alloy (SMA) is introduced into the SD oscillator to form the SMA-SD oscillator to adjust the smoothing parameters. Combining the SMA-SD oscillator with MRF, a nonlinear dynamic vibration absorber (DVA) with variable frequency and damping is designed. The structure and control principle of the designed DVA is studied to achieve the two variable characteristics simultaneously by adjusting the current intensity. Numerical results on a two-degree-of-freedom coupled system show that the proposed DVA can adapt to different working conditions only by adjusting the current intensity

New force transmissibility and optimization for a nonlinear dynamic vibration absorber

Combining the nonlinear characteristic of the smooth and discontinuous (SD) oscillator and the magnetorheological fluid (MRF), a nonlinear dynamic vibration absorber (DVA) with variable frequency and variable damping is constructed. To evaluate its performance, new force transmissibility based on the accessibility and wide applicability of the acceleration is proposed. Specifically, the root mean square is employed to define the force transmissibility as the ratio of the inertial force of the main system and the external force. Furthermore, with the proposed transmissibility as the evaluation index, a two-step optimization method is designed to optimize the parameters of the nonlinear DVA in a wide frequency range, which overcomes the limitation of the traditional single optimization method in broadband vibration reduction. The time-domain dynamic analysis of the main structure is carried out to show the effectiveness and superiority of the two-step optimization method using the time history diagram, phase diagram, Poincaré map, and frequency spectrum. It is worthwhile noting that the two-step optimization method makes it possible to obtain great broadband damping with fewer parameter adjustments, which leads to extend the life of the nonlinear system

Periodic Solution of the Strongly Nonlinear Asymmetry System with the Dynamic Frequency Method

In this article, we present a new accurate iterative and asymptotic method to construct analytical periodic solutions for a strongly nonlinear system, even if it is not Z2-symmetric. This method is applicable not only to a conservative system but also to a non-conservative system with a limit cycle response. Distinct from the general harmonic balance method, it depends on balancing a few trigonometric terms (at most five terms) in the energy equation of the nonlinear system. According to this iterative approach, the dynamic frequency is a trigonometric function that varies with time t, which represents the influence of derivatives of the higher harmonic terms in a compact form and leads to a significant reduction of calculation workload. Two examples were solved and numerical solutions are presented to illustrate the effectiveness and convenience of the method. Based on the present method, we also outline a modified energy balance method to further simplify the procedure of higher order computation. Finally, a nonlinear strength index is introduced to automatically identify the strength of nonlinearity and classify the suitable strategies

Bifurcations and chaotic thresholds for the spring-pendulum oscillator with irrational and fractional nonlinear restoring forces

Nonlinear dynamical systems with irrational and fractional nonlinear restoring forces often occur in both science and engineering, and always lead to a barrier for conventional nonlinear techniques. In this paper, we have investigated the global bifurcations and the chaos directly for a nonlinear system with irrational and fractional nonlinear restoring forces avoiding the conventional Taylor’s expansion to retain the natural characteristics of the system. By introducing a particular dimensionless representation and a series of transformations, the two-degree-of-freedom system can be transformed into a perturbed Hamiltonian system. The extended Melnikov method is directly used to detect the chaotic threshold of the perturbed system theoretically, which overcomes the barrier caused by solving theoretical solution for the homoclinic orbit of the unperturbed system. The numerical simulations are carried out to demonstrate the complicated dynamics of the nonlinear spring-pendulum system, which show the efficiency of the criteria for chaotic motion in the system.<br/