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.

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.

PHASE SYNCHRONIZATION OF PERIODIC AND CHAOTIC STATES INDUCED BY EXTERNAL OPTICAL INJECTION IN SEMICONDUCTOR LASERS

2000 ◽  
Vol 10 (10) ◽  
pp. 2441-2446 ◽  
Author(s):  
E. LARIONTSEV
Keyword(s):  
Numerical Simulation ◽  
Periodic Solutions ◽  
Semiconductor Lasers ◽  
Phase Synchronization ◽  
Period Doubling ◽  
Injection Locking ◽  
Optical Injection ◽  
Frequency Detuning ◽  
Stable Periodic ◽  
Unstable Part

We present results on a numerical simulation of phase synchronization of the periodic and chaotic states induced by optical injection in a semiconductor laser. Such states occur in an unstable part of the injection locking region. We find two branches of the stable periodic solutions (bistability) which are synchronized by the injected signal. We show that phase synchronization of the periodic states exists in the truncated period-doubling cascade to chaos and is maintained in some region of the chaotic states. We present a mapping of the region of synchronized chaotic states as a function of the amplitude and frequency detuning of the external optical injection.

INTERMITTENCY AND CHAOS NEAR HOPF BIFURCATION WITH BROKEN 𝕆(2) × 𝕆(2) SYMMETRY

2013 ◽  
Vol 23 (08) ◽  
pp. 1350139
Author(s):  
YANG ZOU ◽  
GERHARD DANGELMAYR ◽  
IULIANA OPREA
Keyword(s):  
Hopf Bifurcation ◽  
Phase Space ◽  
Normal Form ◽  
Periodic Solutions ◽  
Periodic Orbits ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Coupled System ◽  
Dimensional Model ◽  
Low Dimensional

The eight-dimensional normal form for a Hopf bifurcation with 𝕆(2) × 𝕆(2) symmetry is perturbed by imperfection terms that break a continuous translation symmetry. The parameters of the fully symmetric normal form are fixed to values for which all basic periodic solutions residing in two-dimensional fixed point subspaces are unstable, and the dynamics is attracted by a chaotic attractor resulting from a period doubling cascade of periodic orbits. By using symmetry-adapted variables, the dimension of the phase space of the normal form is reduced to four and the dimension of the perturbed normal form is reduced to five. In the reduced phase space, periodic solutions are revealed as fixed points, and quasiperiodic solutions as periodic orbits. For the perturbed normal form, parameter regimes with different types of chaotic dynamics are identified when the imperfection parameter is varied. The characteristics of this complex dynamics are symmetry breaking and increasing, various period doubling cascades, intermittency and crises, and switching between symmetry-conjugated chaotic saddles. In particular, the perturbed system serves as a low dimensional model for the complicated switching dynamics found in simulations of the globally coupled system of Ginzburg–Landau equations extending the 𝕆(2) × 𝕆(2)-symmetric normal form to account for spatial modulations. In addition, this system can be considered as a low dimensional model for the dynamics of perturbed waves in anisotropic systems with imperfect geometries due to the presence of sidewalls.

scholarly journals On a periodically forced, weakly damped pendulum. Part 2: Horizontal forcing

1990 ◽  
Vol 32 (1) ◽  
pp. 23-41 ◽  
Author(s):  
Peter J. Bryant ◽  
John W. Miles
Keyword(s):  
Angular Velocity ◽  
Periodic Motion ◽  
Numerical Solutions ◽  
Period Doubling ◽  
Dimensionless Frequency ◽  
Periodic Forcing ◽  
Maximum Acceleration ◽  
Oscillatory Solutions ◽  
Stability Boundaries ◽  
Wide Range

AbstractWe consider the phase-locked solutions of the differential equation governing planar motion of a weakly damped pendulum driven by horizontal, periodic forcing of the pivot with maximum acceleration εg and dimensionless frequency ω. Analytical solutions for symmetric oscillations at smaller values of ε are continued into numerical solutions at larger values of ε. A wide range of stable oscillatory solutions is described, including motion that is symmetric or asymmetric, downward or inverted, and at periods equal to the forcing period T ≡ 2π/ω or integral multiples thereof. Stable running oscillations with mean angular velocity pω/q, where p and q are integers, are investigated also. Stability boundaries are calculated for swinging oscillations of period T, 2T and 4T; 3T and 6T; and for running oscillations with mean angular velocity ω. The period-doubling cascades typically culminate in nearly periodic motion followed by chaotic motion or some independent periodic motion.

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.

Periodic orbits and chaos around two black holes

1990 ◽  
Vol 431 (1881) ◽  
pp. 183-202 ◽  
Keyword(s):  
Black Holes ◽  
Periodic Orbits ◽  
Initial Conditions ◽  
Classical Problem ◽  
Period Doubling ◽  
Satellite Orbits ◽  
Unstable Periodic Orbits ◽  
Newtonian Case ◽  
Stable Periodic ◽  
Escaping Orbits

We calculate orbits of photons and particles in the relativistic problem of two extreme Reissner-Nordström black holes (fixed). In the case of photons there are three types of non-periodic orbits, namely orbits falling into the black holes M 1 and M 2 (types I and II), and orbits escaping to infinity (III). The various types of orbits are separated by orbits asymptotic to the three main types of (unstable) periodic orbits: ( a ) around M 1 , ( b ) around M 2 and ( c ) around both M 1 and M 2 . Between two non-periodic orbits of two different types there are orbits of the third type. The initial conditions of the three types of orbits form three Cantor sets, and this fact is a manifestation of chaos. The role of higher-order periodic orbits is explained. In the case of particles the situation is similar in the parabolic and hyperbolic cases. However, in the elliptic case the orbits are contained inside a curve of zero velocity, and there are no escaping orbits. Instead we have orbits trapped around stable periodic orbits (IV) and stochastic orbits not falling on the black holes M 1 and M 2 (V). The stable periodic orbits become unstable by period doubling, and infinite period doublings lead to chaos. The newtonian limit is an integrable problem (essentially the same as the classical problem of two fixed centres). The periodic orbits of type ( c ) are topologically similar to the corresponding relativistic orbits, but they differ considerably numerically. We prove that in the newtonian case there are no satellite orbits around M 1 or M 2 (of type ( a ) or ( b )). The post-newtonian case is non-integrable. In this case there are in general orbits of all three types ( a ), ( b ) and ( c ).

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.

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