Bursting Oscillations and the Mechanism with Sliding Bifurcations in a Filippov Dynamical System

2018 ◽  
Vol 28 (12) ◽  
pp. 1850146 ◽  
Author(s):  
Rui Qu ◽  
Yu Wang ◽  
Guoqing Wu ◽  
Zhengdi Zhang ◽  
Qinsheng Bi
Keyword(s):  
Dynamical System ◽  
Natural Frequency ◽  
Stable Equilibrium ◽  
Multiple Scales ◽  
Hopf Bifurcations ◽  
Periodic Excitation ◽  
Nonsmooth Boundary ◽  
Bursting Oscillations ◽  
First Case ◽  
Exciting Frequency

The main purpose of the paper is to investigate the effect of multiple scales in frequency domain on the complicated oscillations of Filippov system with discontinuous right-hand side. A relatively simple model based on the Chua’s circuit with periodic excitation is introduced as an example. When the exciting frequency is far less than the natural frequency, implying that an order gap between the exciting frequency and the natural frequency exists, the whole exciting term can be considered as a slow-varying parameter, based on which the bifurcations of the two subsystems in different regions divided by the nonsmooth boundary are presented. Two typical cases are considered, which correspond to different distributions of equilibrium branches as well as the related bifurcations. In the first case, periodic symmetric Hopf/Hopf-fold-sliding bursting oscillations can be obtained, in which Hopf bifurcations may cause the alternations between the quiescent states and the spiking states, while fold bifurcations connect the two quiescent states moving along the stable equilibrium branches and sliding along the nonsmooth boundary, respectively. While the second case is the periodic symmetric fold/fold-fold-sliding bursting, where the fold bifurcations not only lead to the alternations between the quiescent states and the spiking states, but also connect the two quiescent states moving along the stable equilibrium branches and sliding along the nonsmooth boundary, respectively. It is pointed out that, different from the bursting oscillations in smooth dynamical systems in which the bifurcations may cause the alternations between quiescent states and spiking states, in the nonsmooth system, bifurcations may not only lead to the alternations, but also connect different forms of quiescent states. Furthermore, in the Filippov system, sliding movement along the nonsmooth boundary can be observed, the mechanism of which is presented based on the analysis of the two subsystems in different regions.

scholarly journals Bursting Oscillations in Shimizu-Morioka System with Slow-Varying Periodic Excitation

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Xindong Ma ◽  
Shuqian Cao
Keyword(s):  
Natural Frequency ◽  
Coupling Effect ◽  
Excitation Frequency ◽  
Time Interval ◽  
Periodic Excitation ◽  
Bursting Oscillations ◽  
Fold Bifurcation ◽  
Exciting Frequency ◽  
Bursting Oscillation ◽  
Parameter Condition

The coupling effect of two different frequency scales between the exciting frequency and the natural frequency of the Shimizu-Morioka system with slow-varying periodic excitation is investigated. First, based on the analysis of the equilibrium states, homoclinic bifurcation, fold bifurcation, and supercritical Hopf bifurcation are observed in the system under a certain parameter condition. When the exciting frequency is much smaller than the natural frequency, we can regard the periodic excitation as a slow-varying parameter. Second, complicated dynamic behaviors are analyzed when the slow-varying parameter passes through different bifurcation points, of which the mechanisms of four different bursting patterns, namely, symmetric “homoclinic/homoclinic” bursting oscillation, symmetric “fold/Hopf” bursting oscillation, symmetric “fold/fold” bursting oscillation, and symmetric “Hopf/Hopf” bursting oscillation via “fold/fold” hysteresis loop, are revealed with different values of the parameterbby means of the transformed phase portrait. Finally, we can find that the time interval between two symmetric adjacent spikes of bursting oscillations exhibits dependency on the periodic excitation frequency.

Slow–Fast Behaviors and Their Mechanism in a Periodically Excited Dynamical System with Double Hopf Bifurcations

2021 ◽  
Vol 31 (08) ◽  
pp. 2130022
Author(s):  
Miaorong Zhang ◽  
Xiaofang Zhang ◽  
Qinsheng Bi
Keyword(s):  
Dynamical System ◽  
Hopf Bifurcation ◽  
Frequency Domain ◽  
Autonomous System ◽  
Single Mode ◽  
Phase Portraits ◽  
Hopf Bifurcations ◽  
Double Hopf Bifurcation ◽  
Exciting Frequency ◽  
Slow Passage

This paper focuses on the influence of two scales in the frequency domain on the behaviors of a typical dynamical system with a double Hopf bifurcation. By introducing an external periodic excitation to the normal form of the vector field with double Hopf bifurcation at the origin and taking the exciting frequency far less than the natural frequency, a theoretical model with two scales in the frequency domain is established. Regarding the whole exciting term as a slow-varying parameter leads to a generalized autonomous system, in which the equilibrium branches and their bifurcations with the variation of the slow-varying parameter can be derived. With the increase of the exciting amplitude, different types of bifurcations may be involved in the generalized autonomous system, resulting in several qualitatively different forms of bursting attractors, the mechanism of which is presented by overlapping the transformed phase portraits and the bifurcations of the equilibrium branches. It is found that the single mode 2D torus may evolve to the bursting attractors with mixed modes, in which the trajectory alternates between the single mode oscillations and the mixed mode oscillations. Furthermore, the transitions between the quiescent states and the spiking states may not occur exactly at the bifurcation points because of the slow passage effect, while Hopf bifurcations may cause different forms of repetitive spiking oscillations.

Fast–Slow Dynamics and Bifurcation Mechanism in a Novel Chaotic System

2019 ◽  
Vol 29 (10) ◽  
pp. 1930028
Author(s):  
Lan Huang ◽  
Guoqing Wu ◽  
Zhengdi Zhang ◽  
Qinsheng Bi
Keyword(s):  
Natural Frequency ◽  
Chaotic System ◽  
Three Dimensional ◽  
Harmonic Excitation ◽  
Phase Portraits ◽  
Fast Subsystem ◽  
Bursting Oscillations ◽  
Exciting Frequency ◽  
Different Types ◽  
Multiple Coexisting Attractors

This paper proposes a novel three-dimensional chaotic system with multiple coexisting attractors, where different values of a constant control parameter may drive the chaotic behaviors to evolve from single-scroll to double-scroll attractors. When the controlling term is replaced by a periodic harmonic excitation where the exciting frequency is far less than the natural frequency, chaotic movement may disappear, while periodic bursting oscillations will take place. Based on the fact that during a period defined by the natural frequency, the exciting term keeps almost a constant, the whole exciting term can be regarded as a slow-varying parameter resulting in a generalized autonomous system, its equilibrium branches as well as the related bifurcations occurring with the variation of the slow-varying parameter are derived. With the increase of the exciting amplitude, asymmetric and symmetric bursting attractors can be observed, for which the mechanism can be analyzed by the overlap of the equilibrium branches and the transformed phase portraits. With different values of the exciting amplitude corresponding to the change region of the slow-varying parameter, different bifurcations such as fold and Hopf bifurcations may involve the bursting structures, leading to different types of bursting oscillations. Furthermore, the phase space can be divided into two regions by a line boundary because of the symmetry of the vector field. When the trajectory from one region returning to the region arrives at the boundary, two asymmetric bursting attractors located in different regions coexist, which are symmetric to each other. However, when the trajectory passes across the boundary, an enlarged symmetric bursting attractor can be observed, whose trajectory connects the two original asymmetric attractors. Furthermore, it is found that when the trajectory runs along a stable equilibrium branch to the bifurcation point, it may move almost strictly along an unstable equilibrium branch of the fast subsystem because of the delay influence of the bifurcation.

scholarly journals Influence of Time Delay on Controlling the Non-Linear Oscillations of a Rotating Blade

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 85
Author(s):  
Yasser Salah Hamed ◽  
Ali Kandil
Keyword(s):  
Time Delay ◽  
Natural Frequency ◽  
Multiple Scales ◽  
Control Process ◽  
Control Force ◽  
Specific Level ◽  
Loop Control ◽  
Positive Displacement ◽  
Non Linear ◽  
Linear Vibrations

Time delay is an obstacle in the way of actively controlling non-linear vibrations. In this paper, a rotating blade’s non-linear oscillations are reduced via a time-delayed non-linear saturation controller (NSC). This controller is excited by a positive displacement signal measured from the sensors on the blade, and its output is the suitable control force applied onto the actuators on the blade driving it to the desired minimum vibratory level. Based on the saturation phenomenon, the blade vibrations can be saturated at a specific level while the rest of the energy is transferred to the controller. This can be done by adjusting the controller natural frequency to be one half of the blade natural frequency. The whole behavior is governed by a system of first-order differential equations gained by the method of multiple scales. Different responses are included to show the influences of time delay on the closed-loop control process. Also, a good agreement can be noticed between the analytical curves and the numerically simulated ones.

scholarly journals Theoretical and Experimental Investigation of Two-to-One Internal Resonance in MEMS Arch Resonators

2018 ◽  
Vol 14 (1) ◽  
Author(s):  
Feras K. Alfosail ◽  
Amal Z. Hajjaj ◽  
Mohammad I. Younis
Keyword(s):  
Internal Resonance ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Hopf Bifurcations ◽  
Chaotic Motions ◽  
Different Types ◽  
Quadratic Nonlinearities ◽  
Multiple Scales Perturbation Method ◽  
Good Agreement ◽  
Perturbation Solutions

We investigate theoretically and experimentally the two-to-one internal resonance in micromachined arch beams, which are electrothermally tuned and electrostatically driven. By applying an electrothermal voltage across the arch, the ratio between its first two symmetric modes is tuned to two. We model the nonlinear response of the arch beam during the two-to-one internal resonance using the multiple scales perturbation method. The perturbation solution is expanded up to three orders considering the influence of the quadratic nonlinearities, cubic nonlinearities, and the two simultaneous excitations at higher AC voltages. The perturbation solutions are compared to those obtained from a multimode Galerkin procedure and to experimental data based on deliberately fabricated Silicon arch beam. Good agreement is found among the results. Results indicate that the system exhibits different types of bifurcations, such as saddle node and Hopf bifurcations, which can lead to quasi-periodic and potentially chaotic motions.

Tunable Wave Propagation in Granular Crystals by Altering Lattice Network Topology

2016 ◽  
Vol 139 (1) ◽  
Author(s):  
Raj Kumar Pal ◽  
Robert F. Waymel ◽  
Philippe H. Geubelle ◽  
John Lambros
Keyword(s):  
Wave Propagation ◽  
Network Topology ◽  
Stable Equilibrium ◽  
Primary Chain ◽  
First Case ◽  
Granular Crystals ◽  
Potential Applications ◽  
Wave Tailoring ◽  
Lattice Network ◽  
Granular Crystal

We develop a framework for wave tailoring by altering the lattice network topology of a granular crystal consisting of spherical granules in contact. The lattice topology can alternate between two stable configurations, with the spherical granules of the lattice held in stable equilibrium in each configuration by gravity. Under impact, the first configuration results in a wave with rapidly decaying amplitude as it propagates along a primary chain, while the second configuration results in a solitary wave propagating along the primary chain with no decay. The mechanism to achieve such tunability is by having energy diverted to the granules adjacent to the primary chain in the first case but not the second. The tunable design of the proposed network is validated using both numerical simulations and experiments. In terms of potential applications, the proposed bistable lattice network can be viewed either as a wave attenuator or as a device that allows higher amplitude wave propagation in one direction than in the opposite direction. The lattice is analogous to a crystal phase transformation due to the change in atomic configurations, leading to the change in properties at the macroscale.

An Experimental and Theoretical Investigation of Nonlinear Vibrations of Piezoelectrically-Driven Microcantilevers

2006 ◽  
Author(s):  
S. Nima Mahmoodi ◽  
Nader Jalili
Keyword(s):  
Natural Frequency ◽  
Piezoelectric Layer ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Galerkin Approximation ◽  
Closed Form Solution ◽  
Cubic Form ◽  
Dynamic Representation ◽  
Microcantilever Beam

The nonlinear vibrations of a piezoelectrically-driven microcantilever beam are experimentally and theoretically investigated. A part of the microcantilever beam surface is covered by a piezoelectric layer, which acts as an actuator. Practically, the first resonance of the beam is of interest, and hence, the microcantilever beam is modeled to obtain the natural frequency theoretically. The bending vibrations of the beam are studied considering the inextensibility condition and the coupling between electrical and mechanical properties in piezoelectric materials. The nonlinear term appears in the form of quadratic due to presence of piezoelectric layer, and cubic form due to geometry of the beam (mainly due to the beam's inextensibility). Galerkin approximation is utilized to discretize the equations of motion. The obtained equation is simulated to find the natural frequency of the system. In addition, method of multiple scales is applied to the equations of motion to arrive at the closed-form solution for natural frequency of the system. The experimental results verify the theoretical findings very closely. It is, therefore, concluded that the nonlinear approach could provide better dynamic representation of the microcantilever than previous linear models.

scholarly journals Cosmological dynamics of f(R) models in dynamical system analysis

2020 ◽  
Vol 35 (22) ◽  
pp. 2050124
Author(s):  
Parth Shah ◽  
Gauranga C. Samanta
Keyword(s):  
Dynamical System ◽  
Asymptotic Behavior ◽  
Critical Points ◽  
System Analysis ◽  
Late Time ◽  
Field Equations ◽  
Acceleration Phase ◽  
First Case ◽  
Acceleration Of The Universe ◽  
The Universe

In this work we try to understand the late-time acceleration of the universe by assuming some modification in the geometry of the space and using dynamical system analysis. This technique allows to understand the behavior of the universe without analytically solving the field equations. We study the acceleration phase of the universe and stability properties of the critical points which could be compared with observational results. We consider an asymptotic behavior of two particular models [Formula: see text] and [Formula: see text] with [Formula: see text], [Formula: see text], [Formula: see text] for the study. As a first case we fix the value of [Formula: see text] and analyze for all [Formula: see text]. Later as second case, we fix the value of [Formula: see text] and calculation are done for all [Formula: see text]. At the end all the calculations for the generalized case have been shown and results have been discussed in detail.

Nonlinear Dynamics of a Taut String with Material Nonlinearities

2000 ◽  
Vol 123 (1) ◽  
pp. 53-60 ◽  
Author(s):  
M. J. Leamy ◽  
O. Gottlieb
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamic Response ◽  
Natural Frequency ◽  
Continuum Mechanics ◽  
Finite Deformation ◽  
Multiple Scales ◽  
String Model ◽  
Taut String ◽  
Linear Material ◽  
Nonlinear String

A spatial string model incorporating a nonlinear (and nonconservative) material law is proposed using finite deformation continuum mechanics. The resulting model is shown to reduce to the classical nonlinear string when a linear material law is used. The influence of material nonlinearities on the string’s dynamic response to excitation near a transverse natural frequency is shown to be small due to their appearance at high orders only. Material nonlinearities appear at low order in the equations for excitation near a longitudinal natural frequency, and a solution for this case is developed by applying a second order multiple scales method directly to the partial differential equations. The material nonlinearities are found to influence both the degree of nonlinearity in the response and its softening or hardening nature.

Vibration Control of Offshore Platforms Using a Wideband METMD System

2006 ◽  
Author(s):  
Dong Zhao ◽  
Rujian Ma ◽  
Dongmei Cai
Keyword(s):  
Vibration Control ◽  
Natural Frequency ◽  
Fem Simulation ◽  
Average Frequency ◽  
Offshore Platforms ◽  
Frequency Bandwidth ◽  
Wave Load ◽  
Control Effect ◽  
Exciting Frequency ◽  
Specific Frequency

A wideband multiple extended tuned mass dampers (METMD) system has been developed for reducing the multiple resonant responses of the platforms to all kinds of loads, such as earthquake, typhoon, tsunami and big ice load. This system is composed of several subsystems, each of which consists of one set of extended tuned mass damper (ETMD) unit covering a specific frequency bandwidth, and its average frequency is tuned to one of the first resonant frequencies of the platform. The offshore platform is simplified to a single degree-of-freedom (DOF) system to which a METMD subsystem (composed of m ETMDs) is attached and constitutes m+1 DOFs system. The total mass ratio of the METMD subsystem to the platform is 14% and the frequency ratio of the exciting frequency to the platform’s natural frequency varies in [0.5, 1.5]. The theory analysis shows that: 1) the platform has the better vibration control effect when the non-dimensional frequency bandwidth Ω, which is defined as the ratio of the frequency range to the controlled (target) platforms natural frequency, is in [0.35, 0.6]; 2) the damping coefficient ξ of ETMD systems is in [0.05, 0.15] and 3) the number of the ETMDs is 5 when Ω = 0.45 and ξ = 0.1. The FEM simulation shows that the METMD has a better vibration control effect on the mega-platforms’ vibration control under the random ocean wave load.

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