scholarly journals Multiple solutions and transient chaos in a nonlinear flexible coupling model

2021 ◽  
Vol 43 (10) ◽  
Author(s):  
Jerzy Margielewicz ◽  
Damian Gąska ◽  
Tadeusz Opasiak ◽  
Grzegorz Litak
Keyword(s):  
Periodic Solutions ◽  
Multiple Solutions ◽  
Computer Modelling ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Coupling Model ◽  
Basins Of Attraction ◽  
Largest Lyapunov Exponent ◽  
Transient Chaos ◽  
Stable Periodic

AbstractThis paper investigates the nonlinear dynamics of a flexible tyre coupling via computer modelling and simulation. The research mainly focused on identifying basins of attraction of coexisting solutions of the formulated phenomenological coupling model. On the basis of the derived mathematical model, and by assuming ranges of variability of the control parameters, the areas in which chaotic clutch movement takes place are determined. To identify multiple solutions, a new diagram of solutions (DS) was used, illustrating the number of coexisting solutions and their periodicity. The DS diagram was drawn based on the fixed points of the Poincaré section. To verify the proposed method of identifying periodic solutions, the graphic image of the DS was compared to the three-dimensional distribution of the largest Lyapunov exponent and the bifurcation diagram. For selected values of the control parameter ω, coexisting periodic solutions were identified, and basins of attraction were plotted. Basins of attraction were determined in relation to examples of coexistence of periodic solutions and transient chaos. Areas of initial conditions that correspond to the phenomenon of unstable chaos are mixed with the conditions of a stable periodic solution, to which the transient chaos is attracted. In the graphic images of the basins of attraction, the areas corresponding to the transient and periodic chaos are blurred.

scholarly journals Subsumed Homoclinic Connections and Infinitely Many Coexisting Attractors in Piecewise-Linear Maps

2017 ◽  
Vol 27 (02) ◽  
pp. 1730010 ◽  
Author(s):  
David J. W. Simpson ◽  
Christopher P. Tuffley
Keyword(s):  
Periodic Solutions ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Stable And Unstable Manifolds ◽  
Homoclinic Connections ◽  
Linear Maps ◽  
Continuous Maps ◽  
Piecewise Linear Maps ◽  
Stable Periodic ◽  
Unstable Manifolds

We establish an equivalence between infinitely many asymptotically stable periodic solutions and subsumed homoclinic connections for [Formula: see text]-dimensional piecewise-linear continuous maps. These features arise as a codimension-three phenomenon. The periodic solutions are single-round: they each involve one excursion away from a central saddle-type periodic solution. The homoclinic connection is subsumed in the sense that one branch of the unstable manifold of the saddle solution is contained entirely within its stable manifold. The results are proved by using exact expressions for the periodic solutions and components of the stable and unstable manifolds which are available because the maps are piecewise-linear. We also describe a practical approach for finding this phenomenon in the parameter space of a map and illustrate the results with the three-dimensional border-collision normal form.

scholarly journals Chaotic breakdown of a periodically forced, weakly damped pendulum

1992 ◽  
Vol 34 (2) ◽  
pp. 153-173 ◽  
Author(s):  
Peter J. Bryant
Keyword(s):  
Phase Space ◽  
Periodic Solutions ◽  
Initial Conditions ◽  
Period Doubling ◽  
Dimensionless Frequency ◽  
Frequency Range ◽  
Time Intervals ◽  
Maximum Acceleration ◽  
Periodic Oscillations ◽  
Stable Periodic

AbstractAn investigation is made of the transition from periodic solutions through nearly-periodic solutions to chaotic solutions of the differential equation governing forced coplanar motion of a weakly damped pendulum. The pendulum is driven by horizontal, periodic forcing of the pivot with maximum acceleration Є g and dimensionless frequency ω As the forcing frequency ω is decreased gradually at a sufficiently large forcing amplitude Є, it has been shown previously that the pendulum progresses from symmetric oscillations of period T (= 2 π/ω) into a symmetry-breaking, period-doubling sequence of stable, periodic oscillations. There are two related forms of asymmetric, stable oscillations in the sequence, dependent on the initial conditions. When the frequency is decreased immediately beyond the sequence, the oscillations become unstable but remain in the neighbourhood in (θ,) phase space of one or other of the two forms of periodic oscillations, where θ(t) is the pendulum angle with the downward vertical. As the frequency is decreased further, the oscillations move intermittently between the neighbourhoods in (θ,) phase space of each of the two forms of periodic oscillations, in paired nearly-periodic oscillations. Further decrease of the forcing frequency leads to time intervals in which the motion is strongly unstable, with the pendulum passing intermittently over the pivot, interspersed with time intervals when the motion is nearly-periodic and only weakly unstable. The strongly-unstable intervals dominate in fully chaotic oscillations. Windows of independent, stable, periodic oscillations occur throughout the frequency range investigated. It is shown in an appendix how the Floquet method may be interpreted to describe the linear stability of the periodic and nearly-periodic solutions, and the windows of periodic oscillations in the investigated frequency range are listed in a second appendix.

HOPF BIFURCATION FROM LINES OF EQUILIBRIA WITHOUT PARAMETERS IN MEMRISTOR OSCILLATORS

2010 ◽  
Vol 20 (02) ◽  
pp. 437-450 ◽  
Author(s):  
MARCELO MESSIAS ◽  
CRISTIANE NESPOLI ◽  
VANESSA A. BOTTA
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Fixed Set ◽  
Electronic Element ◽  
Stable Periodic ◽  
Parameter Values

The memristor is supposed to be the fourth fundamental electronic element in addition to the well-known resistor, inductor and capacitor. Named as a contraction for memory resistor, its theoretical existence was postulated in 1971 by L. O. Chua, based on symmetrical and logical properties observed in some electronic circuits. On the other hand its physical realization was announced only recently in a paper published on May 2008 issue of Nature by a research team from Hewlett–Packard Company. In this work, we present the bifurcation analysis of two memristor oscillators mathematical models, given by three-dimensional five-parameter piecewise-linear and cubic systems of ordinary differential equations. We show that depending on the parameter values, the systems may present the coexistence of both infinitely many stable periodic orbits and stable equilibrium points. The periodic orbits arise from the change in local stability of equilibrium points on a line of equilibria, for a fixed set of parameter values. This phenomenon is a kind of Hopf bifurcation without parameters. We have numerical evidences that such stable periodic orbits form an invariant surface, which is an attractor of the systems solutions. The results obtained imply that even for a fixed set of parameters the two systems studied may or may not present oscillations, depending on the initial condition considered in the phase space. Moreover, when they exist, the amplitude of the oscillations also depends on the initial conditions.

scholarly journals Transient Chaos in Platform-vibrator with Shock

2021 ◽  
pp. 22-40
Author(s):  
Viktor Bazhenov ◽  
Olha Pogorelova ◽  
Tetiana Postnikova
Keyword(s):  
Initial Conditions ◽  
Precast Concrete ◽  
Large Mass ◽  
Contact Forces ◽  
Hertzian Contact ◽  
Upper Body ◽  
Largest Lyapunov Exponent ◽  
Transient Chaos ◽  
Impact System ◽  
Concrete Production

Platform-vibrator with shock is widely used in the construction industry for compacting and molding large concrete products. Its mathematical model, created in our previous work, meets all the basic requirements of shock-vibration technology for the precast concrete production on low-frequency resonant platform-vibrators. This model corresponds to the two-body 2-DOF vibro-impact system with a soft impact. It is strongly nonlinear non-smooth discontinuous system. This is unusual vibro-impact system due to its specific properties. The upper body, with a very large mass, breaks away from the lower body a very short distance, and then falls down onto the soft constraint that causes a soft impact. Then it bounces and falls again, and so on. A soft impact is simulated with nonlinear Hertzian contact force. This model exhibited many unique phenomena inherent in nonlinear non-smooth dynamical systems with varying control parameters. In this paper, we demonstrate the transient chaos in a vibro-impact system. Our finding of transient chaos in platform-vibrator with shock, besides being a remarkable phenomenon by itself, provides an understanding of the dynamical processes that occur in the platform-vibrator when varying the technological mass of the mold with concrete. Phase trajectories, Poincaré maps, graphs of time series and contact forces, Fourier spectra, the largest Lyapunov exponent, and wavelet characteristics are used in numerical investigations to determine the chaotic and periodic phases of the realization. We show both the dependence of the transient chaos on the control parameter value and the sensitive dependence on the initial conditions. We hope that this analysis can help avoid undesirable platform-vibrator behaviour during design and operation due to inappropriate system parameters, since transient chaos may be a dangerous and unwanted state of a vibro-impact system.

Design and FPGA implementation of a memristor-based multi-scroll hyperchaotic system

2022 ◽  
Author(s):  
Sheng-Hao Jia ◽  
Yu-Xia Li ◽  
Qing-Yu Shi ◽  
Xia Huang
Keyword(s):  
Chaotic System ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Nonlinear Function ◽  
Digital Circuit ◽  
Basins Of Attraction ◽  
Hyperchaotic System ◽  
Digital Implementation ◽  
Dynamical Characteristics ◽  
Novel Method

Abstract In this paper, a novel memristor-based multi-scroll hyperchaotic system is proposed. Based on a voltage-controlled memristor and a modulating sine nonlinear function, a novel method is proposed to generate the multi-scroll hyperchaotic attractors. First, a multi-scroll chaotic system is constructed from a three-dimensional chaotic system by designing a modulating sine nonlinear function. Then, a voltage-controlled memristor is introduced into the above-designed multi-scroll chaotic system. Thus, a memristor-based multi-scroll hyperchaotic system is generated, and this hyperchaotic system can produce various coexisting hyperchaotic attractors with different topological structures. Moreover, different number of scrolls and different topological attractors can be obtained by varying the initial conditions of this system without changing the system parameters. The Lyapunov exponents, bifurcation diagrams and basins of attraction are given to analyze the dynamical characteristics of the multi-scroll hyperchaotic system. Besides, the FPGA-based digital implementation of the memristor-based multi-scroll hyperchaotic system is carried out. The experimental results of the FPGA-based digital circuit are displayed on the oscilloscope.

Global Dynamics of a Cooperative Discrete System in the Plane

2018 ◽  
Vol 28 (07) ◽  
pp. 1830022
Author(s):  
Arzu Bilgin ◽  
Ann Brett ◽  
Mustafa R. S. Kulenović ◽  
Esmir Pilav
Keyword(s):  
Periodic Solutions ◽  
Discrete System ◽  
Allee Effect ◽  
Stable Equilibrium ◽  
Initial Conditions ◽  
Global Dynamics ◽  
Cooperative System ◽  
Stable Periodic

In this paper, we consider the cooperative system [Formula: see text] where all parameters [Formula: see text] are positive numbers and the initial conditions [Formula: see text] are nonnegative numbers. We describe the global dynamics of this system in a number of cases. An interesting feature of this system is that it exhibits a coexistence of locally stable equilibrium and locally stable periodic solutions as well as the Allee effect.

scholarly journals Multiple solutions of asymmetric potential bistable energy harvesters: numerical simulation and experimental validation

2019 ◽  
Author(s):  
Chris Bowen
Keyword(s):  
Multiple Solutions ◽  
Initial Conditions ◽  
High Energy ◽  
Negative Influence ◽  
Basins Of Attraction ◽  
Potential Well ◽  
Average Output ◽  
Asymmetric System ◽  
Energy Harvesters ◽  
Output Performance

In this paper we investigate the multiple solutions of nonlinear asymmetric potentialbistable energy harvesters (BEHs) under harmonic excitations. Basins of attraction under certainexcitations explain the existance of multiple solutions in the asymmetric potential BEHs and indicate that the asymmetric system has a higher probability to oscillate in the deeper potential well under low and moderate excitation levels. Thus, the appearance of asymmetric potentials in BEHs has a negative influence on the output performance. Average output powers under different excitation requencies and initial conditions illustrate that the asymmetric potential BEHs are more likely to achieve high-energy branch (HEB) with initial conditions in the shallower potential well, and the probability is influenced by the degree of asymmetry of the BEHs. Finally, experiments are carriedout, and results under constant and sweep frequency excitations demonstrate that the output performance will be actually improved for the asymmetric potential BEHs if the initial oscillations e shallower potential well

scholarly journals On the equivalence of three-dimensional differential systems

2020 ◽  
Vol 18 (1) ◽  
pp. 1164-1172
Author(s):  
Jian Zhou ◽  
Shiyin Zhao
Keyword(s):  
Periodic Solutions ◽  
Differential System ◽  
Necessary Conditions ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Periodic Systems ◽  
Structural Form ◽  
Differential Systems

Abstract In this paper, firstly, we study the structural form of reflective integral for a given system. Then the sufficient conditions are obtained to ensure there exists the reflective integral with these structured form for such system. Secondly, we discuss the necessary conditions for the equivalence of such systems and a general three-dimensional differential system. And then, we apply the obtained results to the study of the behavior of their periodic solutions when such systems are periodic systems in t.

Study of Rehabilitation of Injured Knee Joint Applying Chaotic Theory in Human Body Motion

2007 ◽  
Vol 342-343 ◽  
pp. 581-584
Author(s):  
Byung Young Moon ◽  
Kwon Son ◽  
Jung Hong Park
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Angular Displacement ◽  
Three Dimensional ◽  
Gait Pattern ◽  
Body Motion ◽  
Largest Lyapunov Exponent ◽  
Chaos Analysis ◽  
Nonlinear Technique ◽  
Injured Knee

Gait analysis is essential to identify accurate cause and knee condition from patients who display abnormal walking. Traditional linear tools can, however, mask the true structure of motor variability, since biomechanical data from a few strides during the gait have limitation to understanding the system. Therefore, it is necessary to propose a more precise dynamic method. The chaos analysis, a nonlinear technique, focuses on understanding how variations in the gait pattern change over time. Healthy eight subjects walked on a treadmill for 100 seconds at 60 Hz. Three dimensional walking kinematic data were obtained using two cameras and KWON3D motion analyzer. The largest Lyapunov exponent from the measured knee angular displacement time series was calculated to quantify local stability. This study quantified the variability present in time series generated from gait parameter via chaos analysis. Gait pattern is found to be chaotic. The proposed Lyapunov exponent can be used in rehabilitation and diagnosis of recoverable patients.

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