On the dynamics of rigid-block motion under harmonic forcing

1989 ◽  
Vol 425 (1869) ◽  
pp. 441-476 ◽  
Keyword(s):  
Initial Conditions ◽  
Period Doubling ◽  
Orbital Stability ◽  
Domain Of Attraction ◽  
Rigid Block ◽  
Analogue Circuit ◽  
Safety Issues ◽  
Harmonic Forcing ◽  
Chaotic Nature ◽  
Carry Over

In this paper the simplest and most widely used model of a rigid block undergoing harmonic forcing is analysed in detail. The block is shown to possess extremely complicated dynamics, with many different types of response being revealed. Symmetric single-impact subharmonic orbits of all orders are found and regions of parameter space in which they occur are given. In particular, period-doubling cascades of asymmetric orbits are found, which ultimately produce an apparently non-periodic or chaotic response. Sensitivity to initial conditions is illustrated, which leads to uncertainty in the prediction of the asymptotic dynamics. Nevertheless, the transient response may be the most important in connection with real earthquakes. To this end, the concept of the domain of maximum transients is introduced. In this light the response is shown to be quite ordered and predictable, despite the chaotic nature of the asymptotic domain of attraction. It is shown that safety issues cannot be satisfactorily resolved until an agreed set of initial conditions is established. It appears that blocks may survive under very high accelerations and topple at very low accelerations provided the initial conditions are correct. Consideration is also given to the use of actual earthquake recordings in attempting to reproduce responses in given structures. If the present conclusions carry over to general excitations, then small errors in recordings may produce large differences in response. The present methods include orbital stability techniques together with detailed numerical computations. These results are backed up by encouraging qualitative agreement from an electronic analogue circuit.

On the motion of a rigid block, tethered at one corner, under harmonic forcing

1992 ◽  
Vol 439 (1905) ◽  
pp. 35-45 ◽  
Keyword(s):  
Period Doubling ◽  
Rigid Block ◽  
Harmonic Forcing ◽  
Wide Range ◽  
Side Walls ◽  
Multiple Impact ◽  
Digital Simulations ◽  
First Time ◽  
Subharmonic Orbits ◽  
Period Doubling Bifurcations

A rigid block, tethered at one corner, is subjected to harmonic forcing. The motion is shown to be equivalent to that of the inverted pendulum impacting one of a pair of asymmetrically placed side-walls. The dynamics of the problem contain subharmonic responses, multiple solutions, period-doubling bifurcations, etc. Stability boundaries are given for a wide range of parameters and orbits are shown to be possible for a large range of forcing amplitudes. Some orbits are possible at forcing amplitudes larger than those in the untethered case. A period- and impact-doubling sequence is shown explicitly for the first time, using digital simulations. Evidence is offered for the existence of more than one type of multiple-impact solution. Large amplitude subharmonic orbits are found.

scholarly journals Formation of supermassive black hole seeds in nuclear star clusters via gas accretion and runaway collisions

2021 ◽  
Author(s):  
Arpan Das ◽  
Dominik R G Schleicher ◽  
Nathan W C Leigh ◽  
Tjarda C N Boekholt
Keyword(s):  
Black Holes ◽  
Initial Conditions ◽  
High Redshift ◽  
Star Clusters ◽  
Gas Reservoir ◽  
Highly Sensitive ◽  
Stellar Collisions ◽  
Key Variables ◽  
Chaotic Nature ◽  
Gas Accretion

Abstract More than two hundred supermassive black holes (SMBHs) of masses ≳ 109 M⊙ have been discovered at z ≳ 6. One promising pathway for the formation of SMBHs is through the collapse of supermassive stars (SMSs) with masses ∼103 − 5 M⊙ into seed black holes which could grow upto few times 109 M⊙ SMBHs observed at z ∼ 7. In this paper, we explore how SMSs with masses ∼103 − 5 M⊙ could be formed via gas accretion and runaway stellar collisions in high-redshift, metal-poor nuclear star clusters (NSCs) using idealised N-body simulations. We explore physically motivated accretion scenarios, e.g. Bondi-Hoyle-Lyttleton accretion and Eddington accretion, as well as simplified scenarios such as constant accretions. While gas is present, the accretion timescale remains considerably shorter than the timescale for collisions with the most massive object (MMO). However, overall the timescale for collisions between any two stars in the cluster can become comparable or shorter than the accretion timescale, hence collisions still play a crucial role in determining the final mass of the SMSs. We find that the problem is highly sensitive to the initial conditions and our assumed recipe for the accretion, due to the highly chaotic nature of the problem. The key variables that determine the mass growth mechanism are the mass of the MMO and the gas reservoir that is available for the accretion. Depending on different conditions, SMSs of masses ∼103 − 5 M⊙ can form for all three accretion scenarios considered in this work.

Behavior of a Self-Sustained Electromechanical Transducer and Routes to Chaos

2005 ◽  
Vol 128 (3) ◽  
pp. 282-293 ◽  
Author(s):  
J. C. Chedjou ◽  
K. Kyamakya ◽  
I. Moussa ◽  
H.-P. Kuchenbecker ◽  
W. Mathis
Keyword(s):  
Electronic Circuit ◽  
Initial Conditions ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Electromechanical System ◽  
Phase Portraits ◽  
Coupling Coefficients ◽  
Oscillatory Solutions ◽  
Electromechanical Transducer ◽  
Design Engineers

This paper studies the dynamics of a self-sustained electromechanical transducer. The stability of fixed points in the linear response is examined. Their local bifurcations are investigated and different types of bifurcation likely to occur are found. Conditions for the occurrence of Hopf bifurcations are derived. Harmonic oscillatory solutions are obtained in both nonresonant and resonant cases. Their stability is analyzed in the resonant case. Various bifurcation diagrams associated to the largest one-dimensional (1-D) numerical Lyapunov exponent are obtained, and it is found that chaos can appear suddenly, through period doubling, period adding, or torus breakdown. The extreme sensitivity of the electromechanical system to both initial conditions and tiny variations of the coupling coefficients is also outlined. The experimental study of̱the electromechanical system is carried out. An appropriate electronic circuit (analog simulator) is proposed for the investigation of the dynamical behavior of the electromechanical system. Correspondences are established between the coefficients of the electromechanical system model and the components of the electronic circuit. Harmonic oscillatory solutions and phase portraits are obtained experimentally. One of the most important contributions of this work is to provide a set of reliable analytical expressions (formulas) describing the electromechanical system behavior. These formulas are of great importance for design engineers as they can be used to predict the states of the electromechanical systems and respectively to avoid their destruction. The reliability of the analytical formulas is demonstrated by the very good agreement with the results obtained by both the numeric and the experimental analysis.

A Fuzzy Nonlinear Modeling of a McPherson Suspension System

2005 ◽  
Author(s):  
Ali Tavassoli ◽  
Hamed Jafarian ◽  
Mohammad Eghtesad
Keyword(s):  
Dynamical Systems ◽  
Initial Conditions ◽  
Nonlinear Modeling ◽  
Fuzzy Model ◽  
Domain Of Attraction ◽  
Suspension System ◽  
Reliable Model ◽  
Sugeno Model ◽  
Takagi Sugeno

The Takagi-Sugeno fuzzy model (TSfm) is a universal approximation of continuous real functions that are defined in a closed and bounded subset of Rn. This strong property of the TSfm can find several applications in modeling of dynamical systems that are described by differential equations. In this paper, we consider Takagi-Sugeno fuzzy model for a McPherson suspension system. One advantage of TSfm is its wide domain of attraction in compare with the other methods. To apply TSf modeling, one must precisely choose the nonlinear terms of the system governing equations. For each nonlinear term, there should be selected some linear subsystems that together represent the equivalent of the original nonlinear suspension system. This equivalence, for our case study, is illustrated by simulation results for various road disturbances and initial conditions which show the Takagi-Sugeno model can give a realistic and reliable model for dynamical systems.

Nonlinear Oscillations of a Particle in the Plane Under Longitudinal End Excitation

2003 ◽  
Author(s):  
Rajesh K. Jha ◽  
Robert G. Parker
Keyword(s):  
Nonlinear Oscillations ◽  
Low Frequency ◽  
Period Doubling ◽  
Lumped Parameter Model ◽  
Lumped Parameter ◽  
Parametric Resonances ◽  
Jump Phenomena ◽  
Route To Chaos ◽  
Chaotic Nature ◽  
Two Degree Of Freedom

We study the forced vibrations of a two degree of freedom lumped parameter model of a belt span under longitudinal excitation. The belt inertia is modelled as a particle and the belt elasticity is modelled by two identical linear springs. Numerical integration is used to calculate free responses and perform frequency and amplitude sweeps. Frequency sweep results indicate parametric resonances, jump phenomena, sub- and super-harmonic responses, quasiperiodicity and chaos. Amplitude sweep at a low frequency shows bifurcations of limit cycles and the period doubling route to chaos. Poincare sections are computed to show the chaotic nature of the responses.

scholarly journals RePLaT-Chaos: A Simple Educational Application to Discover the Chaotic Nature of Atmospheric Advection

Atmosphere ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 29
Author(s):  
Tímea Haszpra
Keyword(s):  
Large Scale ◽  
Volcanic Ash ◽  
Initial Conditions ◽  
Industrial Accidents ◽  
Fractal Structures ◽  
Atmospheric Pollutant ◽  
Folded Structure ◽  
Exponential Stretching ◽  
Irregular Motion ◽  
Chaotic Nature

Large-scale atmospheric pollutant spreading via volcano eruptions and industrial accidents may have serious effects on our life. However, many students and non-experts are generally not aware of the fact that pollutant clouds do not disperse in the atmosphere like dye blobs on clothes. Rather, an initially compact pollutant cloud soon becomes strongly stretched with filamentary and folded structure. This is the result of the chaotic behaviour of advection of pollutants in 3-D flows, i.e., the advection dynamics of pollutants shows the typical characteristics such as sensitivity to the initial conditions, irregular motion, and complicated but well-organized (fractal) structures. This study presents possible applications of a software called RePLaT-Chaos by means of which the characteristics of the long-range atmospheric spreading of volcanic ash clouds and other pollutants can be investigated in an easy and interactive way. This application is also a suitable tool for studying the chaotic features of the advection and determines two quantities which describe the chaoticity of the advection processes: the stretching rate quantifies the strength of the exponential stretching of pollutant clouds; and the escape rate characterizes the rate of the rapidity by which the settling particles of a pollutant cloud leave the atmosphere.

scholarly journals Rapid, accurate, precise and reproducible ligand–protein binding free energy prediction

2020 ◽  
Vol 10 (6) ◽  
pp. 20200007 ◽  
Author(s):  
Shunzhou Wan ◽  
Agastya P. Bhati ◽  
Stefan J. Zasada ◽  
Peter V. Coveney
Keyword(s):  
Free Energy ◽  
Initial Conditions ◽  
Binding Free Energy ◽  
Computational Cost ◽  
Free Energy Calculations ◽  
Enhanced Sampling ◽  
Energy Prediction ◽  
Massive Scale ◽  
Chaotic Nature ◽  
High Level

A central quantity of interest in molecular biology and medicine is the free energy of binding of a molecule to a target biomacromolecule. Until recently, the accurate prediction of binding affinity had been widely regarded as out of reach of theoretical methods owing to the lack of reproducibility of the available methods, not to mention their complexity, computational cost and time-consuming procedures. The lack of reproducibility stems primarily from the chaotic nature of classical molecular dynamics (MD) and the associated extreme sensitivity of trajectories to their initial conditions. Here, we review computational approaches for both relative and absolute binding free energy calculations, and illustrate their application to a diverse set of ligands bound to a range of proteins with immediate relevance in a number of medical domains. We focus on ensemble-based methods which are essential in order to compute statistically robust results, including two we have recently developed, namely thermodynamic integration with enhanced sampling and enhanced sampling of MD with an approximation of continuum solvent. Together, these form a set of rapid, accurate, precise and reproducible free energy methods. They can be used in real-world problems such as hit-to-lead and lead optimization stages in drug discovery, and in personalized medicine. These applications show that individual binding affinities equipped with uncertainty quantification may be computed in a few hours on a massive scale given access to suitable high-end computing resources and workflow automation. A high level of accuracy can be achieved using these approaches.

scholarly journals The Chaotic Motion of the Solar System

1993 ◽  
Vol 132 ◽  
pp. 21-21
Author(s):  
J. Laskar
Keyword(s):  
Lyapunov Exponent ◽  
Solar System ◽  
Initial Conditions ◽  
Maximum Lyapunov Exponent ◽  
Secular Resonance ◽  
Critical Argument ◽  
Outer Planets ◽  
Fundamental Frequencies ◽  
Chaotic Nature ◽  
Evolution With Time

AbstractIn a previous paper (Laskar, Nature, 338, 237-238), the chaotic nature of the solar system excluding Pluto was established by the numerical computation of the maximum Lyapunov exponent of its secular system over 200 Myr. In the present an explanation is given for the exponential divergence of the orbits: it is due to the transition from libration to circulation of the critical argument of the secular resonance 2(g4−g3)−(s4−s3) related to the motions of perihelions and nodes of the Birth and Mars. An other important secular resonance is identified: (g1−g5)−(s1−s2). Its critical argument stays in libration over 200 Myr with a period of about 10 Myr and amplitude from 85° to 135°. The main features of the solutions of the inner planets are now identified when taking these resonances into account. Estimates of the size of the chaotic regions are determined by a new numerical method using the evolution with time of the fundamental frequencies. The size of the chaotic regions in the inner solar system are large and correspond to variations of about 0.2 arcsec/year in the fundamental frequencies. The chaotic nature of the inner solar system can thus be considered as robust against small variations of the initial conditions or of the model. The chaotic regions related to the outer planets frequencies are very thin except for g6 which present variations sufficiently large to be significant over the age of the solar system.

scholarly journals Updated studies on exomoons in the HD 23079 system

2021 ◽  
Vol 38 ◽  
Author(s):  
O. Jagtap ◽  
B. Quarles ◽  
M. Cuntz
Keyword(s):  
Initial Conditions ◽  
Time Lag ◽  
Orbital Stability ◽  
Stability Limit ◽  
Outer Edge ◽  
Future Research ◽  
Tidal Interactions ◽  
System A ◽  
Solar Type ◽  
Outward Migration

Abstract We re-evaluate the outer edge of orbital stability for possible exomoons orbiting the radial velocity planet discovered in the HD 23079 system. In this system, a solar-type star hosts a Jupiter-mass planet in a nearly circular orbit in the outer stellar habitable zone. The outer stability limit of exomoons is deduced using N-body and tidal migration simulations considering a large range of initial conditions, encompassing both prograde and retrograde orbits. In particular, we extend previous works by evaluating many values in the satellite mean anomaly to identify and exclude regions of quasi-stability. Future observations of this system can make use of our results through a scale factor relative to the currently measured minimum mass. Using a constant time lag tidal model (Hut 1981), we find that plausible tidal interactions within the system are insufficient to induce significant outward migration toward the theoretical stability limit. While current technologies are incapable of detecting exomoons in this system, we comment on the detectability of putative moons through Doppler monitoring within direct imaging observations in view of future research capacities.

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