Complex Dynamics in a Harmonically Excited Lennard-Jones Oscillator: Microcantilever-Sample Interaction in Scanning Probe Microscopes1

1998 ◽  
Vol 122 (1) ◽  
pp. 240-245 ◽  
Author(s):  
M. Basso ◽  
L. Giarre´ ◽  
M. Dahleh ◽  
I. Mezic´
Keyword(s):  
Parameter Space ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Operating Conditions ◽  
Invariant Sets ◽  
Jones Potential ◽  
Lennard Jones ◽  
Atomic Force

In this paper we model the microcantilever-sample interaction in an atomic force microscope (AFM) via a Lennard-Jones potential and consider the dynamical behavior of a harmonically forced system. Using nonlinear analysis techniques on attracting limit sets, we numerically verify the presence of chaotic invariant sets. The chaotic behavior appears to be generated via a cascade of period doubling, whose occurrence has been studied as a function of the system parameters. As expected, the chaotic attractors are obtained for values of parameters predicted by Melnikov theory. Moreover, the numerical analysis can be fruitfully employed to analyze the region of the parameter space where no theoretical information on the presence of a chaotic invariant set is available. In addition to explaining the experimentally observed chaotic behavior, this analysis can be useful in finding a controller that stabilizes the system on a nonchaotic trajectory. The analysis can also be used to change the AFM operating conditions to a region of the parameter space where regular motion is ensured. [S0022-0434(00)01401-5]

Nonlinear Dynamic Analysis and Chaotic Behavior in Atomic Force Microscopy

2005 ◽  
Author(s):  
Hossein Nejat Pishkenari ◽  
Nader Jalili ◽  
Aria Alasty ◽  
Ali Meghdari
Keyword(s):  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Nonlinear Dynamic Analysis ◽  
Melnikov Method ◽  
Single Mode ◽  
Intermolecular Forces ◽  
Invariant Sets ◽  
Nonlinear Dynamical ◽  
Atomic Force ◽  
Irregular Motion

The atomic force microscope (AFM) system has evolved into a useful tool for direct measurements of intermolecular forces with atomic-resolution characterization that can be employed in a broad spectrum of applications. In this paper, the nonlinear dynamical behavior of the AFM is studied. This is achieved by modeling the microcantilever as a single mode approximation (lumped-parameters model) and considering the interaction between the sample and cantilever in the form of van der Waals potential. The resultant nonlinear system is then analyzed using Melnikov method, which predicts the regions in which only periodic and quasi-periodic motions exist, and also predicts the regions that chaotic motion is possible. Numerical simulations are used to verify the presence of such chaotic invariant sets determined by Melnikov theory. Finally, the amplitude of vibration in which chaos is appeared is investigated and such irregular motion is proven by several methods including Poincare maps, Fourier transform, autocorrelation function and Lyapunov exponents.

CHAOTIC BEHAVIOR OF A PERIODICALLY FORCED PREDATOR–PREY SYSTEM WITH BEDDINGTON–DEANGELIS FUNCTIONAL RESPONSE AND IMPULSIVE PERTURBATIONS

2006 ◽  
Vol 09 (03) ◽  
pp. 209-222 ◽  
Author(s):  
SHUWEN ZHANG ◽  
DEJUN TAN ◽  
LANSUN CHEN
Keyword(s):  
Functional Response ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Periodic Variation ◽  
Periodic Forcing ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Impulsive Perturbations

The effects of periodic forcing and impulsive perturbations on the predator–prey model with Beddington–DeAngelis functional response are investigated. We assume periodic variation in the intrinsic growth rate of the prey as well as periodic constant impulsive immigration of the predator. The dynamical behavior of the system is simulated and bifurcation diagrams are obtained for different parameters. The results show that periodic forcing and impulsive perturbation can very easily give rise to complex dynamics, including quasi-periodic oscillating, a period-doubling cascade, chaos, a period-halving cascade, non-unique dynamics, and period windows.

Bifurcation and Chaos in a Discrete Fractional Order Prey-Predator System Involving Infection in Prey

2020 ◽  
pp. 95-119
Author(s):  
A. George Maria Selvam ◽  
R. Dhineshbabu
Keyword(s):  
Fractional Order ◽  
Control Method ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Order System ◽  
Qualitative Behavior ◽  
Uniqueness Of Solutions ◽  
Predator System

This chapter considers the dynamical behavior of a new form of fractional order three-dimensional continuous time prey-predator system and its discretized counterpart. Existence and uniqueness of solutions is obtained. The dynamic nature of the model is discussed through local stability analysis of the steady states. Qualitative behavior of the model reveals rich and complex dynamics as exhibited by the discrete-time fractional order model. Moreover, the bifurcation theory is applied to investigate the presence of Neimark-Sacker and period-doubling bifurcations at the coexistence steady state taking h as a bifurcation parameter for the discrete fractional order system. Also, the trajectories, phase diagrams, limit cycles, bifurcation diagrams, and chaotic attractors are obtained for biologically meaningful sets of parameter values for the discretized system. Finally, the analytical results are strengthened with appropriate numerical examples and they demonstrate the chaotic behavior over a range of parameters. Chaos control is achieved by the hybrid control method.

Study on the Atomic-Scale Mechanism of Static Friction

2008 ◽  
Author(s):  
Lingyun Ding ◽  
Zhongliang Gong ◽  
Ping Huang
Keyword(s):  
Static Friction ◽  
Atomic Scale ◽  
Interfacial Friction ◽  
Lennard Jones Potential ◽  
Coupled Oscillator ◽  
Jones Potential ◽  
New Model ◽  
Lennard Jones ◽  
Atomic Force ◽  
Coupled Oscillator Model

A new model named as the coupled-oscillator model, is proposed to study the atomic-scale static friction. The Maugis-Dugdal model is used to approximately substitute the Lennard-Jones potential of the interfacial friction in new model. Then, the formulas for static friction force and coefficient calculation are deduced. A comparison between the theoretical result and the experimental value obtained by an atomic force microscope is presented to show the model and the formulas practically feasible.

Experimental Chatter Modeling of Milling Operations

2007 ◽  
Author(s):  
Giuseppe Catania ◽  
Nicolo` Mancinelli
Keyword(s):  
Ad Hoc ◽  
Dynamical Behavior ◽  
Measurement Data ◽  
Period Doubling ◽  
Experimental Tests ◽  
Cutting Parameters ◽  
Operating Conditions ◽  
Depth Of Cut ◽  
Wide Range ◽  
Excited Vibration

This study refers to the investigation on the critical operating condition occurring on high productivity milling machines, known as chatter. This phenomenon is generated by a self-excited vibration, associated with a loss of stability of the system, causing reduced productivity, poor surface finish and noise. This study consists of the theoretical and experimental modeling of machining chatter conditions, in order to develop a real-time monitoring system able to diagnose the occurrence of chatter in advance and to dynamically modify the cutting parameters for process optimization. A prototype NC 3-axis milling machine was ad hoc realized to accomplish this task. The machine was instrumented by a dynamometer table, and a series of accelerometer sensors were mounted in the proximity of the tool spindle and the workpiece. An analytical model was developed, taking into account the periodic cutting force arising during interrupted cutting operation in milling. The system dynamical behavior was described by means of a set of delay differential equations with periodic coefficients. The stability of this system was analyzed by the semi discretization approach based on the Floquet theory. Lobe stability charts were evaluated and associated with frequency diagrams. Two chatter types were analytically and experimentally detected: period-doubling bifurcations and secondary Hopf bifurcations. Measurement data were acquired and analyzed in the time and frequency domain. Several tests were conducted in a wide range of operating conditions, such as radial immersion, depth of cut and spindle speeds and using different tools. Results are reported showing agreement between the numerical analysis and the related experimental tests.

Knaster-like continua and complex dynamics

1993 ◽  
Vol 13 (4) ◽  
pp. 627-634 ◽  
Author(s):  
Robert L. Devaney
Keyword(s):  
Complex Dynamics ◽  
Julia Set ◽  
Dynamical Behavior ◽  
Invariant Sets ◽  
Entire Plane ◽  
Limit Sets

AbstractIn this paper we discuss the topology and dynamics ofEλ(z) = λezwhen λ is real and λ > 1/e. It is known that the Julia set ofEλis the entire plane in this case. Our goal is to show that there are certain natural invariant subsets forEλwhich are topologically Knaster-like continua. Moreover, the dynamical behavior on these invariant sets is quite tame. We show that the only trivial kinds of α- and ω-limit sets are possible.

COLOR MAP OF LYAPUNOV EXPONENTS OF INVARIANT SETS

1999 ◽  
Vol 09 (07) ◽  
pp. 1459-1463 ◽  
Author(s):  
MARCO MONTI ◽  
WILLIAM B. PARDO ◽  
JONATHAN A. WALKENSTEIN ◽  
EPAMINONDAS ROSA ◽  
CELSO GREBOGI
Keyword(s):  
Lyapunov Exponent ◽  
Parameter Space ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Invariant Sets ◽  
Largest Lyapunov Exponent ◽  
Specific Behavior ◽  
Practical Applications ◽  
The Lorenz System ◽  
Different Levels

The largest Lyapunov exponent of the Lorenz system is used as a measure of chaotic behavior to construct parameter space color maps. Each color in these maps corresponds to different values of the Lyapunov exponent and indicates, in parameter space, the locations of different levels of chaos for the Lorenz system. Practical applications of these maps include moving in parameter space from place to place without leaving a region of specific behavior of the system.

Dynamical behavior of nonlinear wave solutions of the generalized Newell–Whitehead–Segel equation

2020 ◽  
Vol 31 (04) ◽  
pp. 2050059
Author(s):  
Asit Saha ◽  
Amiya Das
Keyword(s):  
Nonlinear Wave ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Phase Portraits ◽  
Phase Plane Analysis ◽  
Linear Coefficient ◽  
Wave Solutions ◽  
Nonlinear Wave Solutions

Dynamical behavior of nonlinear wave solutions of the perturbed and unperturbed generalized Newell–Whitehead–Segel (GNWS) equation is studied via analytical and computational approaches for the first time in the literature. Bifurcation of phase portraits of the unperturbed GNWS equation is dispensed using phase plane analysis through symbolic computation and it shows stable oscillation of the traveling waves. Chaotic behavior of the perturbed GNWS equation is obtained by applying different computational tools, like phase plot, time series plot, Poincare section, bifurcation diagram and Lyapunov exponent. A period-doubling bifurcation behavior to chaotic behavior is shown for the perturbed GNWS equation and again it shows chaotic to periodic motion through inverse period-doubling bifurcation. The perturbed GNWS equation also shows chaotic motion through a sequence of periodic motions (period-1, period-3 and period-5) depending on the variation of the parameter of linear coefficient. Thus, the parameter of linear coefficient plays the role of a controlling parameter in the chaotic dynamics of the perturbed GNWS equation.

scholarly journals Standing Swells Surveyed Showing Surprisingly Stable Solutions for the Lorenz '96 Model

2014 ◽  
Vol 24 (10) ◽  
pp. 1430027 ◽  
Author(s):  
Morgan R. Frank ◽  
Lewis Mitchell ◽  
Peter Sheridan Dodds ◽  
Christopher M. Danforth
Keyword(s):  
Parameter Space ◽  
Chaotic Dynamics ◽  
Degrees Of Freedom ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Standing Waves ◽  
Nonlinear Dynamical ◽  
Hidden Order ◽  
Stable Solutions ◽  
Lorenz 96

The Lorenz '96 model is an adjustable dimension system of ODEs exhibiting chaotic behavior representative of the dynamics observed in the Earth's atmosphere. In the present study, we characterize statistical properties of the chaotic dynamics while varying the degrees of freedom and the forcing. Tuning the dimensionality of the system, we find regions of parameter space with surprising stability in the form of standing waves traveling amongst the slow oscillators. The boundaries of these stable regions fluctuate regularly with the number of slow oscillators. These results demonstrate hidden order in the Lorenz '96 system, strengthening the evidence for its role as a hallmark representative of nonlinear dynamical behavior.

scholarly journals Complex dynamics of a three species food-chain model with Holling type IV functional response

2011 ◽  
Vol 16 (3) ◽  
pp. 553-374
Author(s):  
Ranjit Kumar Upadhyay ◽  
Sharada Nandan Raw
Keyword(s):  
Food Chain ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Chain Model ◽  
Narrow Region ◽  
Type Iv ◽  
Numerical Bifurcation ◽  
Parameter Values ◽  
Food Chain Model

In this paper, dynamical complexities of a three species food chain model with Holling type IV predator response is investigated analytically as well as numerically. The local and global stability analysis is carried out. The persistence criterion of the food chain model is obtained. Numerical bifurcation analysis reveals the chaotic behavior in a narrow region of the bifurcation parameter space for biologically realistic parameter values of the model system. Transition to chaotic behavior is established via period-doubling bifurcation and some sequences of distinctive period-halving bifurcation leading to limit cycles are observed.

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