A Dripping Faucet as a Nonextensive System

2004 ◽  
Author(s):  
T. J. P. Penna ◽  
J. C. Sartorelli
Keyword(s):  
Long Range ◽  
Statistical Physics ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Power Laws ◽  
Main Task ◽  
Nonextensive Statistical Mechanics ◽  
System A ◽  
Microscopic Interactions ◽  
Probability Densities

Here we present our attempt to characterize a time series of drop-to-drop intervals from a dripping faucet as a nonextensive system. We found a long-range anticorrelated behavior as evidence of memory in the dynamics of our system. The hypothesis of faucets dripping at the edge of chaos is reinforced by results of the linear rate of the increase of the nonextensive Tsallis statistics. We also present some similarities between dripping faucets and healthy hearts…. Many systems in Nature exhibit complex or chaotic behaviors. Chaotic behavior is characterized by short-range correlations and strong sensitivity to small changes of the initial conditions. Complex behavior is characterized by the presence of long-range power-law correlations in its dynamics. In the latter, the sensitivity to a perturbation of the initial condition is weaker than in the former. Because the probability densities are frequently described as inverse power laws, the variance and the mean often diverge. Although it is hard to predict the long-term behavior of such systems, it is still possible to get some information from them and even to find similarities between two apparently very distinct systems. Tools from statistical physics are frequently used because the main task here is to deal with diverse macroscopic phenomena and to try to explain them, starting with the microscopic interactions among many individual components. The microscopic interactions are not necessarily complicated, but the collective behavior can determine a rather intricate macroscopic description. Nonextensive statistical mechanics, since its proposal in 1988 [27], has been applied to an impressive collection of systems in which spatial or temporal longrange correlations appear. Hence, it can also become a useful tool to characterize such systems. Here, we present an attempt of using such formalism to try to understand the intriguing behavior of an apparently simple system: a dripping faucet.

Some Observations of Chaotic Vibration Phenomena in High-Speed Rotordynamics

1991 ◽  
Vol 113 (1) ◽  
pp. 50-57 ◽  
Author(s):  
F. F. Ehrich
Keyword(s):  
Numerical Model ◽  
High Speed ◽  
Chaotic Behavior ◽  
Narrow Region ◽  
System A ◽  
Vibratory Response ◽  
Local Contact ◽  
Jeffcott Rotor ◽  
Subharmonic Response ◽  
Static Part

Subharmonic response in rotordynamics may be encountered when a rotor is operated with its rotational centerline eccentric to that of a close clearance static part, so that local contact can take place during each orbit when the rotor is excited by residual unbalance. The rotor will tend to bounce at or near its fundamental frequency when the rotor is operated at or near a speed which is a whole number [n] times that frequency. Using a simple numerical model of a Jeffcott rotor mounted on a nonlinear spring, it is found that the vibratory response in the transition zone midway between adjacent zones of subharmonic response has all the characteristics of chaotic behavior. The transition from subharmonic to chaotic response has a complex substructure which involves a sequence of bifurcations of the orbit with variations in speed. This class of rotordynamic behavior was confirmed and illustrated by experimental observations of the vibratory response of a high-speed turbomachine, operating at a speed between 8 and 9 times its fundamental rotor frequency when in local contact across a clearance in the support system. A narrow region between zones of 8th order and 9th order subharmonic response was identified where the response had all the characteristics of the chaotic motion identified in the numerical model.

scholarly journals Dynamic Analysis and Cryptographic Application of a Sinusoidal-polynomial Composite Chaotic System

2021 ◽  
Author(s):  
Hegui Zhu ◽  
Jiangxia Ge ◽  
Wentao Qi ◽  
Xiangde Zhang ◽  
Xiaoxiong Lu
Keyword(s):  
Image Encryption ◽  
Chaotic System ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Encryption Algorithm ◽  
Topological Transitivity ◽  
Detailed Simulation ◽  
Definition Of ◽  
Image Encryption Algorithm ◽  
Sensitivity To Initial Conditions

Abstract Owning to complex properties of ergodicity, non-periodic ability and sensitivity to initial states, chaotic systems are widely used in cryptography. In this paper, we propose a sinusoidal--polynomial composite chaotic system (SPCCS), and prove that it satisfies Devaney's definition of chaos: the sensitivity to initial conditions, topological transitivity and density of periodic points. The experimental results show that the SPCCS has better unpredictability and more complex chaotic behavior than the classical chaotic maps. Furthermore, we provide a new image encryption algorithm combining pixel segmentation operation, block chaotic matrix confusing operation, and pixel diffusion operation with the SPCCS. Detailed simulation results verify effectiveness of the proposed image encryption algorithm.

Control of a legged robot

2015 ◽  
Vol 24 (1-2) ◽  
pp. 53-57
Author(s):  
Maxime Sadre
Keyword(s):  
Reference Model ◽  
Stabilization Energy ◽  
Main Task ◽  
Legged Robot ◽  
Energy Shaping ◽  
Gait Generation ◽  
System A ◽  
Nonlinear Energy

AbstractThis paper deals with the control of hopping and running systems that interact intermittently with the environment. The control, based on a nonlinear energy reference model, has the main task of conferring to the system, a periodic stable behavior. This approach may be used for gait generation, nominal stance stabilization, energy shaping, and optimization.

scholarly journals A Type of Set Membership Estimation Designed for Dynamic Flux Balance Models

Processes ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1762
Author(s):  
Xin Shen ◽  
Hector Budman
Keyword(s):  
State Space ◽  
Initial Conditions ◽  
Variable Structure ◽  
Upper And Lower Bounds ◽  
Structure System ◽  
E Coli ◽  
Flux Balance ◽  
System A ◽  
Variable Structure System ◽  
Set Membership

Dynamic flux balance models (DFBM) are used in this study to infer metabolite concentrations that are difficult to measure online. The concentrations are estimated based on few available measurements. To account for uncertainty in initial conditions the DFBM is converted into a variable structure system based on a multiparametric linear programming (mpLP) where different regions of the state space are described by correspondingly different state space models. Using this variable structure system, a special set membership-based estimation approach is proposed to estimate unmeasured concentrations from few available measurements. For unobservable concentrations, upper and lower bounds are estimated. The proposed set membership estimation was applied to batch fermentation of E. coli based on DFBM.

scholarly journals Climate Vibrations and Chaotic Behavior of Earth’s Inclination in the Laskar’s Model of a Moonless Earth

2019 ◽  
Vol 224 ◽  
pp. 03009
Author(s):  
Tatjana Gurina ◽  
Vyacheslav Salin
Keyword(s):  
Chaotic Behavior ◽  
Initial Conditions ◽  
Problem Parameter ◽  
System Of Differential Equations ◽  
Hamilton Function ◽  
Autonomous Hamiltonian System ◽  
The Earth ◽  
Time Period ◽  
Rotation Of The Earth ◽  
Regular Region

The model of the moonless Earth, introduced by J. Laskar, has the form of a non-autonomous Hamiltonian system of differential equations for two variables: the cosine of the angle of inclination and the longitude of the axis of rotation of the Earth. The system describes the rotational dynamics of the Earth under the influence of the sun and planets. Earth perturbations from other planets of the solar system are considered periodic and are taken into account using the first four terms of the Fourier expansion of the corresponding part of the Hamilton function with known amplitudes and frequencies. The initial inclination of the Earth is considered as a parameter of the problem. The system was numerically integrated over a time period of 18 million years for various values of the initial inclination from 0 to 180 degrees. Three chaotic gaps of the initial inclination were found. During the bifurcation study, singular points were found and special segments of the non-autonomous system were obtained. A bifurcation diagram of the system is constructed by the initial inclination parameter. Poincare cartographic maps are constructed. The system is written in variations on the initial conditions for the Laskar system, and with its help the dependences of the problem parameter of the senior Lyapunov exponent and the averaged MEGNO indicator are calculated. The results confirm the presence of three chaotic and one regular region of variation of the bifurcation parameter of the problem.

scholarly journals Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1803
Author(s):  
Pattrawut Chansangiam
Keyword(s):  
Equilibrium Point ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Initial Point ◽  
Electrical Circuit ◽  
Nonlinear Resistor ◽  
Voltage Current Characteristic ◽  
Time Waveform

This paper investigates the chaotic behavior of a modified jerk circuit with Chua’s diode. The Chua’s diode considered here is a nonlinear resistor having a symmetric piecewise linear voltage-current characteristic. To describe the system, we apply fundamental laws in electrical circuit theory to formulate a mathematical model in terms of a third-order (jerk) nonlinear differential equation, or equivalently, a system of three first-order differential equations. The analysis shows that this system has three collinear equilibrium points. The time waveform and the trajectories about each equilibrium point depend on its associated eigenvalues. We prove that all three equilibrium points are of type saddle focus, meaning that the trajectory of (x(t),y(t)) diverges in a spiral form but z(t) converges to the equilibrium point for any initial point (x(0),y(0),z(0)). Numerical simulation illustrates that the oscillations are dense, have no period, are highly sensitive to initial conditions, and have a chaotic hidden attractor.

CHAOS IN THE REORIENTATION PROCESS OF A DUAL-SPIN SPACECRAFT WITH TIME-DEPENDENT MOMENTS OF INERTIA

2000 ◽  
Vol 10 (05) ◽  
pp. 997-1018 ◽  
Author(s):  
M. IÑARREA ◽  
V. LANCHARES
Keyword(s):  
Chaotic Behavior ◽  
Initial Conditions ◽  
Center Of Mass ◽  
Melnikov Method ◽  
Moments Of Inertia ◽  
Suitable Parameter ◽  
Nutation Angle ◽  
Principal Axes ◽  
Reorientation Process ◽  
Subharmonic Resonances

We study the spin-up dynamics of a dual-spin spacecraft containing one axisymmetric rotor which is parallel to one of the principal axes of the spacecraft. It will be supposed that one of the moments of inertia of the platform is a periodic function of time and that the center of mass of the spacecraft is not modified. Under these assumptions, it is shown that in the absence of external torques and spinning rotors the system possesses chaotic behavior in the sense that it exhibits Smale's horseshoes. We prove this statement by means of the Melnikov method. The presence of chaotic behavior results in a random spin-up operation. This randomness is visualized by means of maps of the initial conditions with final nutation angle close to zero. This phenomenon is well described by a suitable parameter that measures the amount of randomness of the process. Finally, we relate this parameter with the Melnikov function in the absence of the spinning rotor and with the presence of subharmonic resonances.

Disease Spread and Its Effect on Population Dynamics in Heterogeneous Environment

2016 ◽  
Vol 26 (01) ◽  
pp. 1650004 ◽  
Author(s):  
Ranjit Kumar Upadhyay ◽  
Parimita Roy
Keyword(s):  
Temporal Dynamics ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Reaction Diffusion ◽  
Fish Population ◽  
Reaction Diffusion Equations ◽  
Disease Spread ◽  
Spatial Gradient ◽  
Appreciable Effect ◽  
Heterogeneous Environment

In this paper, an eco-epidemiological model in which both species diffuse along a spatial gradient has been shown to exhibit temporal chaos at a fixed point in space. The proposed model is a modification of the model recently presented by Upadhyay and Roy [2014]. The spatial interactions among the species have been represented in the form of reaction–diffusion equations. The model incorporates the intrinsic growth rate of fish population which varies linearly with the depth of water. Numerical results show that diffusion can drive otherwise stable system into aperiodic behavior with sensitivity to initial conditions. We show that spatially induced chaos plays an important role in spatial pattern formation in heterogeneous environment. Spatiotemporal distributions of species have been simulated using the diffusivity assumptions realistic for natural eco-epidemic systems. We found that in heterogeneous environment, the temporal dynamics of both the species are drastically different and show chaotic behavior. It was also found that the instability observed in the model is due to spatial heterogeneity and diffusion-driven. Cumulative death rate of predator has an appreciable effect on model dynamics as the spatial distribution of all constituent populations exhibit significant changes when this model parameter is changed and it acts as a regularizing factor.

scholarly journals Synchronization of Synchronous Reluctance Motors Using the Discrete Sliding Mode Control Technique

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Salahuddin Abdul Rahman ◽  
Mohamed Zribi ◽  
Nejib Smaoui
Keyword(s):  
Sliding Mode Control ◽  
Sliding Mode ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Control Technique ◽  
Mode Control ◽  
Synchronous Reluctance Motor ◽  
Control Scheme ◽  
Bounded Disturbances ◽  
Simulation Results

The synchronous reluctance motor (SynRM) drive system is known to exhibit chaotic behavior under specified conditions. In this paper, the discrete-time sliding mode control (DSMC) technique is used to synchronize two SynRMs starting from different sets of initial conditions. The mixed variable speed reaching law is adopted in the design of the controller scheme. The parameters of the designed control scheme are tuned using a genetic algorithm (GA). Simulation results are presented to demonstrate the effectiveness of the proposed controller. In addition, the performance of the proposed control scheme is studied through simulations when bounded disturbances and mismatches between the parameters of the systems and those of the control scheme exist. The simulation results show that the designed control scheme is robust to bounded external disturbances and to mismatches in the parameters of the systems.

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