scholarly journals Effects of Symmetric and Asymmetric Nonlinearity on the Dynamics of a Third-Order Autonomous Duffing–Holmes Oscillator

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-26
Author(s):  
Isaac Sami Doubla ◽  
Jacques Kengne ◽  
Raoul Blaise Wafo Tekam ◽  
Zeric Tabekoueng Njitacke ◽  
Clotaire Thierry Sanjong Dagang
Keyword(s):  
Cross Sections ◽  
Analog Computer ◽  
Phase Portraits ◽  
Quadratic Term ◽  
Third Order ◽  
Frequency Spectra ◽  
Amplitude Control ◽  
Attraction Basins ◽  
Asymmetric Nonlinearity ◽  
Control And Synchronization

A generalized third-order autonomous Duffing–Holmes system is proposed and deeply investigated. The proposed system is obtained by adding a parametric quadratic term m x 2 to the cubic nonlinear term − x 3 of an existing third-order autonomous Duffing–Holmes system. This modification allows the system to feature smoothly adjustable nonlinearity, symmetry, and nontrivial equilibria. A particular attention is given to the effects of symmetric and asymmetric nonlinearity on the dynamics of the system. For the specific case of m = 0 , the system is symmetric and interesting phenomena are observed, namely, coexistence of symmetric bifurcations, presence of parallel branches, and the coexistence of four (periodic-chaotic) and six (periodic) symmetric attractors. For m ≠ 0 , the system loses its symmetry. This favors the emergence of other behaviors, such as the coexistence of asymmetric bifurcations, involving the coexistence of several asymmetric attractors (periodic-periodic or periodic-chaotic). All these phenomena have been numerically highlighted using nonlinear dynamic tools (bifurcation diagrams, Lyapunov exponents, phase portraits, time series, frequency spectra, Poincaré section, cross sections of the attraction basins, etc.) and an analog computer of the system. In fact, PSpice simulations of the latter confirm numerical results. Moreover, amplitude control and synchronization strategies are also provided in order to promote the exploitation of the proposed system in engineering.

Elastic Waves Created During Tensile Fracture: The Phenomenon of a Second Fracture

1953 ◽  
Vol 20 (1) ◽  
pp. 122-130
Author(s):  
Julius Miklowitz
Keyword(s):  
Cross Sections ◽  
Tensile Tests ◽  
Analog Computer ◽  
Tensile Fracture ◽  
Strain Waves ◽  
Unloading Wave ◽  
Second Fracture ◽  
Wave Compression ◽  
Initial Fracture ◽  
The Moment

Abstract In some tensile tests with brittle materials, it was noted that fractures were produced at two different cross sections of the specimen when the rupture load was reached. The phenomenon of the second fracture prompted the present investigation. It is believed that the second fracture is caused by the destructive action of the elastic strain waves created during the first of the two fractures. The analytical and experimental work carried out was focused on describing the character of these waves. Consideration of the mechanics involved reduces the problem to that of a vibrating cantilever beam with time-dependent boundary conditions. Two types of waves are shown to exist. The first is a longitudinal unloading wave (compression). The other is a group of flexural strain waves caused by the moment that develops at the initial fracture section. The methods of operational mathematics and the electric-analog computer have been employed in the analytical study.

Fatigue Properties of Al-Li Plates Joined by Friction Stir Welding

2008 ◽  
Vol 385-387 ◽  
pp. 849-852 ◽  
Author(s):  
Pasquale Cavaliere ◽  
Francesco W. Panella ◽  
Antonio Squillace
Keyword(s):  
Mechanical Properties ◽  
Cross Sections ◽  
Testing Machine ◽  
Rolling Direction ◽  
Tensile Tests ◽  
Fatigue Tests ◽  
Control Mode ◽  
Fatigue Properties ◽  
Amplitude Control ◽  
Friction Stir

Al-Li alloys are characterized by a strong anisotropy in mechanical properties and microstructure with respect to the rolling direction. Plates of 2198 Al-Li alloy were friction stir welded by employing maximum rotation speed: 1000 rev/min and welding speed of 80 mm/min, both in parallel and orthogonal directions with respect to the rolling one. The joints mechanical properties were evaluated by means of tensile tests at room temperature. In addition, fatigue tests performed with a resonant electro-mechanical testing machine under constant amplitude control up to 250 Hz loading, were conducted in axial control mode with R(σmin/σmax)=0.33, for all the welding and rotating speed conditions. The fatigue crack propagation experiments were performed by employing single edge notched specimens.With the aim to characterize the weld performances, both the microstructure evolution at jointed cross sections, related to the welding variables, and the fractured surfaces were respectively analyzed by means of optical and scanning electron microscopy.

The Geometry of Quadratic Differential Systems with a Weak Focus of Third Order

2004 ◽  
Vol 56 (2) ◽  
pp. 310-343 ◽  
Author(s):  
Jaume Llibre ◽  
Dana Schlomiuk
Keyword(s):  
Limit Cycles ◽  
Quadratic Differential ◽  
Topological Classification ◽  
Phase Portraits ◽  
Third Order ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Weak Focus ◽  
Affine Invariants ◽  
Geometric Concepts

AbstractIn this article we determine the global geometry of the planar quadratic differential systems with a weak focus of third order. This class plays a significant role in the context of Hilbert's 16-th problem. Indeed, all examples of quadratic differential systems with at least four limit cycles, were obtained by perturbing a system in this family. We use the algebro-geometric concepts of divisor and zero-cycle to encode global properties of the systems and to give structure to this class. We give a theorem of topological classification of such systems in terms of integer-valued affine invariants. According to the possible values taken by them in this family we obtain a total of 18 topologically distinct phase portraits. We show that inside the class of all quadratic systems with the topology of the coefficients, there exists a neighborhood of the family of quadratic systems with a weak focus of third order and which may have graphics but no polycycle in the sense of [15] and no limit cycle, such that any quadratic system in this neighborhood has at most four limit cycles.

Torsional Vibration of a Spur Gear System With Time-Varying and Square Nonlinearities

2013 ◽  
Author(s):  
Qian Ding ◽  
Wei Zhang
Keyword(s):  
Torsional Vibration ◽  
Excitation Frequency ◽  
Harmonic Balance Method ◽  
Spur Gear ◽  
Gear System ◽  
Phase Portraits ◽  
Resonant Peak ◽  
Time Varying ◽  
Frequency Spectra ◽  
Input Torque

This paper investigates the torsional vibration of a spur gear system with time-varying and square nonlinearities, by both the analytical method and numerical simulation. First, the equations of motion of a rotating spur gear system are established. Then a single-dof equivalent system is induced to describe the relative motion or torsional vibration of the gears. The harmonic balance method is used to obtain the steady-state response. Influence of the input torque on the response is discussed and a phenomenon, one resonant peak split up into two peaks when the input torque is high enough is revealed. Last, numerical simulations are carried out and bifurcation diagrams and amplitude-frequency curve is given by taking the excitation frequency as control parameter. Selected typical motions are also presented in detail by time-histories, phase portraits, Poincaré map and frequency spectra.

scholarly journals Stability, Bifurcation, and Quenching Chaos of a Vehicle Suspension System

2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Chun-Cheng Chen ◽  
Shun-Chang Chang
Keyword(s):  
Chaotic Motion ◽  
Control Method ◽  
Homoclinic Orbits ◽  
Suspension System ◽  
State Feedback Control ◽  
Phase Portraits ◽  
Frequency Spectra ◽  
The Road ◽  
Dynamics And Control ◽  
Vehicle Suspension System

This study investigated the dynamics and control of a nonlinear suspension system using a quarter-car model that is forced by the road profile. Bifurcation analysis used to characterize nonlinear dynamic behavior revealed codimension-two bifurcation and homoclinic orbits. The nonlinear dynamics were determined using bifurcation diagrams, phase portraits, Poincaré maps, frequency spectra, and Lyapunov exponents. The Lyapunov exponent was used to identify the onset of chaotic motion. Finally, state feedback control was used to prevent chaotic motion. The effectiveness of the proposed control method was determined via numerical simulations.

Coexistence of Chaos with Hyperchaos, Period-3 Doubling Bifurcation, and Transient Chaos in the Hyperchaotic Oscillator with Gyrators

2015 ◽  
Vol 25 (04) ◽  
pp. 1550052 ◽  
Author(s):  
J. Kengne
Keyword(s):  
Integrated Circuit ◽  
Nonlinear Dynamical Systems ◽  
Piecewise Linear ◽  
Phase Portraits ◽  
Piecewise Linear Approximation ◽  
Transient Chaos ◽  
Frequency Spectra ◽  
Nonlinear Dynamical ◽  
System Parameters ◽  
Coexistence Of Attractors

In this paper, the dynamics of the paradigmatic hyperchaotic oscillator with gyrators introduced by Tamasevicius and co-workers (referred to as the TCMNL oscillator hereafter) is considered. This well known hyperchaotic oscillator with active RC realization of inductors is suitable for integrated circuit implementation. Unlike previous literature based on piecewise-linear approximation methods, I derive a new (smooth) mathematical model based on the Shockley diode equation to explore the dynamics of the oscillator. Various tools for detecting chaos including bifurcation diagrams, Lyapunov exponents, frequency spectra, phase portraits and Poincaré sections are exploited to establish the connection between the system parameters and various complex dynamic regimes (e.g. hyperchaos, period-3 doubling bifurcation, coexistence of attractors, transient chaos) of the hyperchaotic oscillator. One of the most interesting and striking features of this oscillator discovered/revealed in this work is the coexistence of a hyperchaotic attractor with a chaotic one over a broad range of system parameters. This phenomenon was not reported previously and therefore represents a meaningful contribution to the understanding of the behavior of nonlinear dynamical systems in general. A close agreement is observed between theoretical and experimental analyses.

The 1 s -2 s excitation of hydrogen atoms by electron impact

1960 ◽  
Vol 258 (1293) ◽  
pp. 245-250 ◽  
Keyword(s):  
Electron Impact ◽  
Cross Section ◽  
Cross Sections ◽  
Born Approximation ◽  
Impact Excitation ◽  
Electron Impact Excitation ◽  
Hydrogen Atoms ◽  
Third Order ◽  
The Third ◽  
Intermediate States

The expression for the cross-section obtained from the second Born approximation by including only terms to the third order in the interaction energy is employed to calculate cross-sections for the electron impact excitation of the 2 s level of atomic hydrogen, allow­ance being made for distortion and polarization due to the 1 s , 2 s and 2 p 0.± 1 intermediate states. These cross-sections are compared with the available experimental data.

Multiscroll Chaotic Sea Obtained from a Simple 3D System Without Equilibrium

2016 ◽  
Vol 26 (02) ◽  
pp. 1650031 ◽  
Author(s):  
Sajad Jafari ◽  
Viet-Thanh Pham ◽  
Tomasz Kapitaniak
Keyword(s):  
Numerical Simulations ◽  
Lyapunov Exponents ◽  
Chaotic Systems ◽  
Poincaré Map ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Equilibrium System ◽  
Frequency Spectra ◽  
Chaotic Flows ◽  
New System

Recently, many rare chaotic systems have been found including chaotic systems with no equilibria. However, it is surprising that such a system can exhibit multiscroll chaotic sea. In this paper, a novel no-equilibrium system with multiscroll hidden chaotic sea is introduced. Besides having multiscroll chaotic sea, this system has two more interesting properties. Firstly, it is conservative (which is a rare feature in three-dimensional chaotic flows) but not Hamiltonian. Secondly, it has a coexisting set of nested tori. There is a hidden torus which coexists with the chaotic sea. This new system is investigated through numerical simulations such as phase portraits, Lyapunov exponents, Poincaré map, and frequency spectra. Furthermore, the feasibility of such a system is verified through circuital implementation.

scholarly journals Study on Nonlinear Dynamics and Chaos Suppression of Active Magnetic Bearing Systems Based on Synchronization

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Shun-Chang Chang
Keyword(s):  
Nonlinear Dynamics ◽  
Chaotic Behavior ◽  
Nonlinear Dynamic Analysis ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Phase Portraits ◽  
Largest Lyapunov Exponent ◽  
Frequency Spectra ◽  
Control Approach ◽  
Chaos Suppression

This study employed a variety of nonlinear dynamic analysis techniques to explore the complex phenomena associated with a nonlinear mathematical model of an active magnetic bearing (AMB) system. The aim was to develop a method with which to assume control over chaotic behavior. The bifurcation diagram comprehensively explicates rich nonlinear dynamics over a range of parameter values. In this study, we examined the complex nonlinear behaviors of AMB systems using phase portraits, Poincaré maps, and frequency spectra. Furthermore, estimates of the largest Lyapunov exponent based on the properties of synchronization confirmed the occurrence of chatter vibration indicative of chaotic motion. Thus, the proposed continuous feedback control approach based on synchronization characteristics eliminates chaotic oscillations. Finally, some simulation results demonstrated the feasibility and efficiency of the proposed control scheme.

THIRD-ORDER NONLINEAR OPTIC AND OPTICAL LIMITING PROPERTIES OF A MN(III) TRANSITION METAL COMPLEX

2007 ◽  
Vol 16 (04) ◽  
pp. 505-518 ◽  
Author(s):  
ASLI KARAKAS ◽  
AYHAN ELMALI ◽  
YASEMIN YAHSI ◽  
HULYA KARA
Keyword(s):  
Transition Metal ◽  
Metal Complex ◽  
Cross Sections ◽  
Transition Metal Complex ◽  
Optical Limiting ◽  
Optical Nonlinearity ◽  
Photon Absorption ◽  
Third Order ◽  
Hartree Fock ◽  
Vis Spectroscopy

N,N′-bis(5-bromosalicylidene)propane-1,2-diamine-O,O′,N,N′)-manganese(III) chloride transition metal complex has been synthesized and characterized by elemental analysis and UV-vis spectroscopy. Its crystal structure has been determined using X-ray diffraction analysis. To provide an insight into the optical limiting (OL) behavior of the title compound, the third-order nonlinear optical (NLO) properties, one-photon absorption (OPA) and two-photon absorption (TPA) characterizations have been theoretically investigated by means of the time-dependent Hartree–Fock (TDHF), AM1 and configuration interaction (CI) methods, respectively. According to ab initio calculation results, the examined molecule exhibits second hyperpolarizabilities (γ) with non-zero values at the positions of TPA peaks, implying microscopic third-order optical nonlinearity. The maximum OPA wavelengths recorded by linear optical experiment and quantum mechanical computations are estimated in the UV region to be shorter than 400 nm, showing good optical transparency to the visible light. The TPA cross-sections (δ(ω)) at [Formula: see text] values indicate that the synthesized compound might possess OL phenomena, which are in accord with the experimental observations on the manganese complexes in the literature.

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