Intermittency Route to Chaos of a Cantilevered Pipe Conveying Fluid With a Mass Defect at the Free End

1995 ◽  
Vol 62 (4) ◽  
pp. 903-907 ◽  
Author(s):  
C. Semler ◽  
M. P. Pai¨doussis
Keyword(s):  
Period Doubling ◽  
Type I ◽  
Pipe Conveying Fluid ◽  
Mass Defect ◽  
Lumped Mass ◽  
Route To Chaos ◽  
Cantilevered Pipe ◽  
Planar Motions ◽  
Period Doubling Bifurcations ◽  
Conveying Fluid

The nonlinear equations for planar motions of a vertical cantilevered pipe conveying fluid are modified to take into account a small lumped mass added at the free end. The resultant equations contain nonlinear inertial terms; by discretizing the system first and inverting the inertia matrix, these terms are transferred into other matrices. In this paper, the dynamics of the system is examined when the added mass is negative (a mass defect), by means of numerical computations and by the software package AUTO. The system loses stability by a Hopf bifurcation, and the resultant limit cycle undergoes pitchfork and period-doubling bifurcations. Subsequently, as shown by the computation of Floquet multipliers, a type I intermittency route to chaos is followed—as illustrated further by a Lorenz return map, revealing the well-known normal form for this type of bifurcation. The period between “turbulent bursts” of nonperiodic oscillations is computed numerically, as well as Lyapunov exponents. Remarkable qualitative agreement, in both cases, is obtained with analytical results.

Chaotic Motions of a Constrained Pipe Conveying Fluid: Comparison Between Simulation, Analysis, and Experiment

1991 ◽  
Vol 58 (2) ◽  
pp. 559-565 ◽  
Author(s):  
M. P. Paidoussis ◽  
G. X. Li ◽  
R. H. Rand
Keyword(s):  
Analytical Model ◽  
Degrees Of Freedom ◽  
Simulation Analysis ◽  
Period Doubling ◽  
Critical Flow ◽  
Pipe Conveying Fluid ◽  
Chaotic Motions ◽  
Critical Flow Velocity ◽  
Period Doubling Bifurcations ◽  
Conveying Fluid

A refined analytical model is presented for the dynamics of a cantilevered pipe conveying fluid and constrained by motion limiting restraints. Calculations with the discretized form of this model with a progressively increasing number of degrees of freedom, N, show that convergence is achieved with N = 4 or 5, which agrees with previously performed fractal dimension calculations of experimental data. Theory shows that, beyond the Hopf bifurcation, as the flow is increased, a pitchfork bifurcation is followed by a cascade of period doubling bifurcations leading to chaos, which is in qualitative agreement with observation. The numerically computed theoretical critical flow velocities are in excellent quantitative agreement (5–10 percent) with experimental values for the thresholds of the Hopf and period doubling bifurcations and for the onset of chaos. An approximation for the critical flow velocity for the loss of stability of the post-Hopf limit cycle is also obtained by using center manifold concepts and normal form techniques for a simplified version of the analytical model; it is found that the values obtained in this manner are approximately within 10 percent of those computed numerically.

Period-Doubling Bifurcation and Non-Typical Route to Chaos of a Two-Degree-Of-Freedom Vibro-Impact System

2000 ◽  
Vol 68 (4) ◽  
pp. 670-674 ◽  
Author(s):  
G. L. Wen and ◽  
J. H. Xie
Keyword(s):  
Finite Number ◽  
Period Doubling ◽  
Degree Of Freedom ◽  
Period Doubling Bifurcation ◽  
Periodic Response ◽  
Impact System ◽  
Route To Chaos ◽  
Two Degree Of Freedom ◽  
Impact Motion ◽  
Period Doubling Bifurcations

A nontypical route to chaos of a two-degree-of-freedom vibro-impact system is investigated. That is, the period-doubling bifurcations, and then the system turns out to the stable quasi-periodic response while the period 4-4 impact motion fails to be stable. Finally, the system converts into chaos through phrase locking of the corresponding four Hopf circles or through a finite number of times of torus-doubling.

CHAOTIC BEHAVIOR INDUCED BY THERMAL MODULATION IN A MODEL OF THE CONVECTIVE FLOW OF A FLUID MIXTURE IN A POROUS MEDIUM

1999 ◽  
Vol 09 (02) ◽  
pp. 383-396 ◽  
Author(s):  
J.-M. MALASOMA ◽  
P. WERNY ◽  
C.-H. LAMARQUE
Keyword(s):  
Porous Medium ◽  
Convective Flow ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Chaotic Attractors ◽  
Global Behavior ◽  
Type I ◽  
Fluid Mixture ◽  
Numerical Investigations ◽  
Period Doubling Bifurcations

Numerical investigations of the global behavior of a model of the convective flow of a binary mixture in a porous medium are reported. We find a complex behavior characterized by the presence of coexisting periodic, quasiperiodic and chaotic attractors. Bifurcations of periodic solutions and routes to chaos via type-I intermittency and period-doubling bifurcations are described. Boundary crises and band merging crises have also been observed.

Routes to Chaos in Squeeze-Film Dampers

2004 ◽  
Author(s):  
Jawaid I. Inayat-Hussain ◽  
Njuki W. Mureithi
Keyword(s):  
Period Doubling ◽  
Squeeze Film ◽  
Saddle Node Bifurcation ◽  
Chaotic Vibrations ◽  
Saddle Node ◽  
Highly Nonlinear ◽  
Squeeze Film Dampers ◽  
Route To Chaos ◽  
Type 3 ◽  
Period Doubling Bifurcations

This work reports on a numerical study undertaken to investigate the imbalance response of a rigid rotor supported by squeeze-film dampers. Two types of damper configurations were considered, namely, dampers without centering springs, and eccentrically operated dampers with centering springs. For a rotor fitted with squeeze-film dampers without centering springs, the study revealed the existence of three regimes of chaotic motion. The route to chaos in the first regime was attributed to a sequence of period-doubling bifurcations of the period-1 (synchronous) rotor response. A period-3 (one-third subharmonic) rotor whirl orbit, which was born from a saddle-node bifurcation, was found to co-exist with the chaotic attractor. The period-3 orbit was also observed to undergo a sequence of period-doubling bifurcations resulting in chaotic vibrations of the rotor. The route to chaos in the third regime of chaotic rotor response, which occurred immediately after the disappearance of the period-3 orbit due to a saddle-node bifurcation, was attributed to a possible boundary crisis. The transitions to chaotic vibrations in the rotor supported by eccentric squeeze-film dampers with centering springs were via the period-doubling cascade and type 3 intermittency routes. The type 3 intermittency transition to chaos was due to an inverse period-doubling bifurcation of the period-2 (one-half subharmonic) rotor response. The unbalance response of the squeeze-film-damper supported rotor presented in this work leads to unique non-synchronous and chaotic vibration signatures. The latter provide some useful insights into the design and development of fault diagnostic tools for rotating machinery that operate in highly nonlinear regimes.

EFFECT OF DISSIPATION ON THE HAMILTONIAN CHAOS IN COUPLED OSCILLATOR SYSTEMS

2002 ◽  
Vol 12 (04) ◽  
pp. 859-867 ◽  
Author(s):  
V. SHEEJA ◽  
M. SABIR
Keyword(s):  
Degrees Of Freedom ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Dissipation Function ◽  
Hamiltonian Chaos ◽  
Fixed Point Attractor ◽  
Point Attractor ◽  
Route To Chaos ◽  
Period Doubling Bifurcations ◽  
External Driving

We study the effect of linear dissipative forces on the chaotic behavior of coupled quartic oscillators with two degrees of freedom. The effect of quadratic Rayleigh dissipation functions, one with diagonal coefficients only and the other with nondiagonal coefficients as well are studied. It is found that the effect of Rayleigh Dissipation function with diagonal coefficients is to suppress chaos in the system and to lead the system to its equilibrium state. However, with a dissipation function with nondiagonal elements, other types of behaviors — including fixed point attractor, periodic attractors and even chaotic attractors — are possible even when there is no external driving. In such a system the route to chaos is through period-doubling bifurcations. This result contradicts the view that linear dissipation always causes decay of oscillations in oscillator models.

Multiple Scenarios of Transition to Chaos in the Alternative Splicing Model

2017 ◽  
Vol 27 (02) ◽  
pp. 1730006 ◽  
Author(s):  
Vladislav V. Kogai ◽  
Vitaly A. Likhoshvai ◽  
Stanislav I. Fadeev ◽  
Tamara M. Khlebodarova
Keyword(s):  
Alternative Splicing ◽  
Period Doubling ◽  
Genetic System ◽  
Protein Isoforms ◽  
Type I ◽  
Transition To Chaos ◽  
Delayed Arguments ◽  
The Mathematical Model ◽  
Period Doubling Bifurcations ◽  
Type Of Transition

We have investigated the scenarios of transition to chaos in the mathematical model of a genetic system constituted by a single transcription factor-encoding gene, the expression of which is self-regulated by a feedback loop that involves protein isoforms. Alternative splicing results in the synthesis of protein isoforms providing opposite regulatory outcomes — activation or repression. The model is represented by a differential equation with two delayed arguments. The possibility of transition to chaos dynamics via all classical scenarios: a cascade of period-doubling bifurcations, quasiperiodicity and type-I, type-II and type-III intermittencies, has been numerically demonstrated. The parametric features of each type of transition to chaos have been described.

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