REGULAR AND CHAOTIC DYNAMICS IN BOUNCING BALL MODELS

2011 ◽  
Vol 21 (03) ◽  
pp. 869-884 ◽  
Author(s):  
SEBASTIAN VOGEL ◽  
STEFAN J. LINZ
Keyword(s):  
Chaotic Dynamics ◽  
Temporal Dynamics ◽  
Initial Conditions ◽  
Coefficient Of Restitution ◽  
Velocity Dependence ◽  
System Parameters ◽  
Bouncing Ball ◽  
Ball Bouncing ◽  
Long Time ◽  
Inelastic Ball

One of the prime paradigms for complex temporal dynamics, the motion of an inelastic ball bouncing on a sinusoidally oscillating table, is revisited. Using extensive numerical simulations, we address the not yet conclusively settled problem of the occurrence of chaos in the partially elastic case. We systematically investigate the spectrum of long-time solutions as function of the initial conditions and system parameters. Subsequently, we generalize the bouncing ball system by taking the velocity dependence of the coefficient of restitution into account and exemplarily demonstrate the drastic impact of such a generalization on the overall dynamics.

Complex Velocity Dependence of the Coefficient of Restitution of a Bouncing Ball

2013 ◽  
Vol 110 (25) ◽  
Author(s):  
Patric Müller ◽  
Michael Heckel ◽  
Achim Sack ◽  
Thorsten Pöschel
Keyword(s):  
Coefficient Of Restitution ◽  
Velocity Dependence ◽  
Complex Velocity ◽  
Bouncing Ball

The Dynamics of Clattering II: Global Results and Shock Protection

1998 ◽  
Vol 120 (1) ◽  
pp. 94-102 ◽  
Author(s):  
S. Goyal ◽  
J. M. Papadopoulos ◽  
P. A. Sullivan
Keyword(s):  
Initial Conditions ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
Maximum Velocity ◽  
Coefficient Of Restitution ◽  
Radius Of Gyration ◽  
Effective Coefficient ◽  
Oblique Angle ◽  
System Parameters ◽  
Shock Protection

The study of “clattering” motion, that occurs when flat objects strike the ground at an oblique angle, is continued. Previously derived equations of motion are solved for values of several clattering quantities (like total number of impacts, maximum velocity jumps in the sequence, time-durations, etc.) for any value of system parameters and initial conditions. The plotted results are examined for their relevance to shock protection. To improve the drop-tolerance of portable products, it is advised that fragile components be located close to the center of mass of the product; that the product be given a low radius of gyration by placing the heavier components, and perhaps even superfluous mass, toward the middle; and that any hard casing be designed (in terms of material and structure) to minimize its effective coefficient of restitution.

scholarly journals Time Dependent Lyotropic Chromonic Textures in Microfluidic Confinements

Crystals ◽  
2020 ◽  
Vol 11 (1) ◽  
pp. 35 ◽  
Author(s):  
Anshul Sharma ◽  
Irvine Lian Hao Ong ◽  
Anupam Sengupta
Keyword(s):  
Phase Transition ◽  
Aspect Ratio ◽  
Temporal Dynamics ◽  
Initial Conditions ◽  
Flow Behavior ◽  
Medical Diagnostics ◽  
Time Dependent ◽  
Disodium Cromoglycate ◽  
M Phase ◽  
Static Conditions

Nematic and columnar phases of lyotropic chromonic liquid crystals (LCLCs) have been long studied for their fundamental and applied prospects in material science and medical diagnostics. LCLC phases represent different self-assembled states of disc-shaped molecules, held together by noncovalent interactions that lead to highly sensitive concentration and temperature dependent properties. Yet, microscale insights into confined LCLCs, specifically in the context of confinement geometry and surface properties, are lacking. Here, we report the emergence of time dependent textures in static disodium cromoglycate (DSCG) solutions, confined in PDMS-based microfluidic devices. We use a combination of soft lithography, surface characterization, and polarized optical imaging to generate and analyze the confinement-induced LCLC textures and demonstrate that over time, herringbone and spherulite textures emerge due to spontaneous nematic (N) to columnar M-phase transition, propagating from the LCLC-PDMS interface into the LCLC bulk. By varying the confinement geometry, anchoring conditions, and the initial DSCG concentration, we can systematically tune the temporal dynamics of the N- to M-phase transition and textural behavior of the confined LCLC. Overall, the time taken to change from nematic to the characteristic M-phase textures decreased as the confinement aspect ratio (width/depth) increased. For a given aspect ratio, the transition to the M-phase was generally faster in degenerate planar confinements, relative to the transition in homeotropic confinements. Since the static molecular states register the initial conditions for LC flows, the time dependent textures reported here suggest that the surface and confinement effects—even under static conditions—could be central in understanding the flow behavior of LCLCs and the associated transport properties of this versatile material.

Evolution of solitary waves in a two-pycnocline system

2009 ◽  
Vol 642 ◽  
pp. 235-277 ◽  
Author(s):  
M. NITSCHE ◽  
P. D. WEIDMAN ◽  
R. GRIMSHAW ◽  
M. GHRIST ◽  
B. FORNBERG
Keyword(s):  
Solitary Waves ◽  
Initial Conditions ◽  
Oscillation Period ◽  
Resonance Phenomenon ◽  
Numerical Code ◽  
Asymptotic Model ◽  
Narrow Region ◽  
Near Resonance ◽  
Long Time ◽  
The Waves

Over two decades ago, some numerical studies and laboratory experiments identified the phenomenon of leapfrogging internal solitary waves located on separated pycnoclines. We revisit this problem to explore the behaviour of the near resonance phenomenon. We have developed a numerical code to follow the long-time inviscid evolution of isolated mode-two disturbances on two separated pycnoclines in a three-layer stratified fluid bounded by rigid horizontal top and bottom walls. We study the dependence of the solution on input system parameters, namely the three fluid densities and the two interface thicknesses, for fixed initial conditions describing isolated mode-two disturbances on each pycnocline. For most parameter values, the initial disturbances separate immediately and evolve into solitary waves, each with a distinct speed. However, in a narrow region of parameter space, the waves pair up and oscillate for some time in leapfrog fashion with a nearly equal average speed. The motion is only quasi-periodic, as each wave loses energy into its respective dispersive tail, which causes the spatial oscillation magnitude and period to increase until the waves eventually separate. We record the separation time, oscillation period and magnitude, and the final amplitudes and celerity of the separated waves as a function of the input parameters, and give evidence that no perfect periodic solutions occur. A simple asymptotic model is developed to aid in interpretation of the numerical results.

On New Aspects of Nested Carbon Nanotubes as Gigahertz Oscillators

2011 ◽  
Vol 133 (5) ◽  
Author(s):  
R. Ansari ◽  
B. Motevalli
Keyword(s):  
Carbon Nanotubes ◽  
Initial Velocity ◽  
Initial Conditions ◽  
Strong Dependence ◽  
Interaction Force ◽  
Arbitrary Length ◽  
Oscillatory Frequency ◽  
Analytical Expression ◽  
System Parameters ◽  
The Core

Nested carbon nanotubes exhibit telescopic oscillatory motion with frequencies in the gigahertz range. In this paper, our previously proposed semi-analytical expression for the interaction force between two concentric carbon nanotubes is used to solve the equation of motion. That expression also enables a new semi-analytical expression for the precise evaluation of oscillation frequency to be introduced. Alternatively, an algebraic frequency formula derived based on the simplifying assumption of constant van der Waals force is also given. Based on the given formulas, a thorough study on different aspects of operating frequencies under various system parameters is conducted, which permits fresh insight into the problem. Some notable improvements over the previously drawn conclusions are made. The strong dependence of oscillatory frequency on system parameters including the extrusion distance and initial velocity of the core as initial conditions for the motion is shown. Interestingly, our results indicate that there is a special initial velocity at which oscillatory frequency is unique for any arbitrary length of the core. A particular relationship between the escape velocity (the minimum initial velocity beyond which the core will leave the outer nanotube) and this specific initial velocity is also revealed.

Analytical Study of the Oblique Impact of a Rod with a Flat Using an Elasto-Plastic Contact Model

2015 ◽  
Vol 801 ◽  
pp. 25-32
Author(s):  
Ozdes Cermik ◽  
Hamid Ghaednia ◽  
Dan B. Marghitu
Keyword(s):  
Impact Velocity ◽  
Contact Force ◽  
Analytical Study ◽  
Initial Conditions ◽  
Permanent Deformation ◽  
Oblique Impact ◽  
Coefficient Of Restitution ◽  
Contact Model ◽  
Impact Angles ◽  
The Impact

In the current study a flattening contact model, combined with a permanent deformation expression, has been analyzed for the oblique impact case. The model has been simulated for different initial conditions using MATLAB. The initial impact velocity used for the simulations ranges from 0.5 to 3 m/s. The results are compared theoretically for four different impact angles including 20, 45, 70, and 90 degrees. The contact force, the linear and the angular motion, the permanent deformation, and the coefficient of restitution have been analyzed. It is assumed that sliding occurs throughout the impact.

Hidden Extreme Multistability, Antimonotonicity and Offset Boosting Control in a Novel Fractional-Order Hyperchaotic System Without Equilibrium

2018 ◽  
Vol 28 (13) ◽  
pp. 1850167 ◽  
Author(s):  
Sen Zhang ◽  
Yicheng Zeng ◽  
Zhijun Li ◽  
Chengyi Zhou
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
State Feedback Control ◽  
Hyperchaotic System ◽  
Hidden Attractors ◽  
System Parameters ◽  
Offset Boosting ◽  
Extreme Multistability ◽  
Hidden Extreme Multistability ◽  
New System

Recently, the notion of hidden extreme multistability and hidden attractors is very attractive in chaos theory and nonlinear dynamics. In this paper, by utilizing a simple state feedback control technique, a novel 4D fractional-order hyperchaotic system is introduced. Of particular interest is that this new system has no equilibrium, which indicates that its attractors are all hidden and thus Shil’nikov method cannot be applied to prove the existence of chaos for lacking hetero-clinic or homo-clinic orbits. Compared with other fractional-order chaotic or hyperchaotic systems, this new system possesses three unique and remarkable features: (i) The amazing and interesting phenomenon of the coexistence of infinitely many hidden attractors with respect to same system parameters and different initial conditions is observed, meaning that hidden extreme multistability arises. (ii) By varying the initial conditions and selecting appropriate system parameters, the striking phenomenon of antimonotonicity is first discovered, especially in such a fractional-order hyperchaotic system without equilibrium. (iii) An attractive special feature of the convenience of offset boosting control of the system is also revealed. The complex and rich hidden dynamic behaviors of this system are investigated by using conventional nonlinear analysis tools, including equilibrium stability, phase portraits, bifurcation diagram, Lyapunov exponents, spectral entropy complexity, and so on. Furthermore, a hardware electronic circuit is designed and implemented. The hardware experimental results and the numerical simulations of the same system on the Matlab platform are well consistent with each other, which demonstrates the feasibility of this new fractional-order hyperchaotic system.

scholarly journals An Efficient Method of Image Encryption Using Rossler Chaotic System

2021 ◽  
Vol 10 (2) ◽  
pp. 11
Author(s):  
Yasir Ahmed Hamza ◽  
Marwan Dahar Omer
Keyword(s):  
Image Encryption ◽  
Chaotic System ◽  
Initial Conditions ◽  
Computation Time ◽  
Encryption Algorithm ◽  
Brute Force ◽  
Symmetric Encryption ◽  
New Approach ◽  
System Parameters ◽  
Plain Image

In this study, a new approach of image encryption has been proposed. This method is depends on the symmetric encryption algorithm RC4 and Rossler chaotic system. Firstly, the encryption key is employed to ciphering a plain image using RC4 and obtains a ciphered-image. Then, the same key is used to generate the initial conditions of the Rossler system. The system parameters and the initial conditions are used as the inputs for Rossler chaotic system to generate the 2-dimensional array of random values. The resulted array is XORed with the ciphered-image to obtain the final encrypted-image. Based on the experimental results, the proposed method has achieved high security and less computation time. Also, the proposed method can be resisted attacks like (statistical, brute-force, and differential).

THE INFLUENCES OF TEMPERATURE AND CHAIN–CHAIN INTERACTION ON FEATURES OF SOLITONS EXCITED IN α-HELIX PROTEIN MOLECULES WITH THREE CHANNELS

2009 ◽  
Vol 23 (10) ◽  
pp. 2303-2322 ◽  
Author(s):  
XIAO-FENG PANG ◽  
MEI-JIE LIU
Keyword(s):  
Energy Transport ◽  
Initial Conditions ◽  
Dynamic Features ◽  
New Model ◽  
Protein Molecules ◽  
Long Time ◽  
The Stability ◽  
Α Helix ◽  
Increasing Temperature ◽  
Chain Interaction

The dynamic features of soliton transporting the bio-energy in the α-helix protein molecules with three channels under influences of temperature of systems and chain–chain interaction among these channels have been numerically studied by using the dynamic equations in a new model and the fourth-order Runge–Kutta method. This result obtained shows that the chain–chain interaction depresses the stability of the soliton due to the dispersed effect, but the stability of the soliton in the case of simultaneous motivation of three channels by an initial conditions is better than that in another initial condition. We also find from this investigation that the new soliton can transport steadily over 1000 amino acid residues in the cases of motion of long time of 120 ps, and retain their shapes and energies to travel towards the protein molecules after mutual collision of the solitons at the biological temperatures of 300 K. Therefore the soliton is very robust against the thermal perturbation of the α-helix protein molecules at 300 K. From the investigation of changes of features of the soliton with increasing temperature, we find that the amplitudes and velocities of the solitons decrease with increasing temperature of proteins, but the soliton disperses in the cases of higher temperature of 325 K and larger structure disorders. Thus we find that the critical temperature of the soliton occurring in the α-helix protein molecules is about 320 K. Therefore we can conclude that the soliton in the new model can play an important role in the bio-energy transport in the α-helix protein molecules with three channels at biological temperature, and the new model is possibly a candidate for the mechanism of this transport.

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