Effects of parametric disorder on a stationary bifurcation

2006 ◽  
Vol 218 (2) ◽  
pp. 139-157 ◽  
Author(s):  
M. Hammele ◽  
S. Schuler ◽  
W. Zimmermann
Keyword(s):  
Stationary Bifurcation

Transitions Through Period Doubling Route to Chaos

1991 ◽  
Author(s):  
R. M. Evan-lwanowski ◽  
Chu-Ho Lu
Keyword(s):  
Period Doubling ◽  
Mirror Image ◽  
Flip Flop ◽  
Physical Systems ◽  
Starting Conditions ◽  
Spatial System ◽  
Stationary Bifurcation ◽  
Extreme Sensitivity ◽  
Route To Chaos ◽  
Parameter Values

Abstract The Duffing driven, damped, “softening” oscillator has been analyzed for transition through period doubling route to chaos. The forcing frequency and amplitude have been varied in time (constant sweep). The stationary 2T, 4T… chaos regions have been determined and used as the starting conditions for nonstationary regimes, consisting of the transition along the Ω(t)=Ω0±α2t,f=const., Ω-line, and along the E-line: Ω(t)=Ω0±α2t;f(t)=f0∓α2t. The results are new, revealing, puzzling and complex. The nonstationary penetration phenomena (delay, memory) has been observed for a single and two-control nonstationary parameters. The rate of penetrations tends to zero with increasing sweeps, delaying thus the nonstationary chaos relative to the stationary chaos by a constant value. A bifurcation discontinuity has been uncovered at the stationary 2T bifurcation: the 2T bifurcation discontinuity drops from the upper branches of (a, Ω) or (a, f) curves to their lower branches. The bifurcation drops occur at the different control parameter values from the response x(t) discontinuities. The stationary bifurcation discontinuities are annihilated in the nonstationary bifurcation cascade to chaos — they reside entirely on the upper or lower nonstationary branches. A puzzling drop (jump) of the chaotic bifurcation bands has been observed for reversed sweeps. Extreme sensitivity of the nonstationary bifurcations to the starting conditions manifests itself in the flip-flop (mirror image) phenomena. The knowledge of the bifurcations allows for accurate reconstruction of the spatial system itself. The results obtained may model mathematically a number of engineering and physical systems.

Bifurcations, symmetries and the notion of fixed subspace

2021 ◽  
pp. 2130006
Author(s):  
Giampaolo Cicogna
Keyword(s):  
Topological Index ◽  
Ad Hoc ◽  
Index Theory ◽  
Typical Case ◽  
One Dimensional ◽  
Dimension Formula ◽  
Symmetry Properties ◽  
Equivariant Bifurcation ◽  
Stationary Bifurcation ◽  
Bifurcation Problems

In the context of stationary bifurcation problems admitting a symmetry, this paper is focused on the key notion of Fixed Subspace (FS), and provides a review of some applications aimed at detecting bifurcating solutions in various situations. We start recalling, in its commonly used simplified version, the old Equivariant Bifurcation Lemma (EBL), where the FS is one-dimensional; then we provide a first generalization in a typical case of non-semisimple critical eigenvalues, where the presence of the symmetry produces a non-trivial situation. Next, we consider the case of FSs of dimension [Formula: see text] in very different contexts. First, relying on the topological index theory and in particular on the Krasnosel’skii theorem, we provide a largely applicable statement of an extension of the EBL. Second, we propose a completely different and new application which combines symmetry properties with the notion of stability of bifurcating solutions. We also provide some simple examples, constructed ad hoc to illustrate the various situations.

Non-Dimensional Parameters Governing the Onset of Wake-Induced Marine Riser Collision

2003 ◽  
Author(s):  
Shan Huang ◽  
Wusheng Wu
Keyword(s):  
Critical Current ◽  
Large Amplitude ◽  
Low Frequency ◽  
Critical Value ◽  
Current Speed ◽  
Loss Of Stability ◽  
Marine Riser ◽  
Dimensional Parameters ◽  
Stationary Bifurcation

Wake-induced riser oscillation can lead to riser clashing. Furthermore, the onset of the wake-induced riser oscillation, which is typified by its large amplitude and low frequency, is attributed to the loss of stability of the downstream riser in the wake once the current exceeds a critical value. The loss of stability is mathematically characterised by a stationary bifurcation. Based upon our previous work, further results are presented in the paper on the non-dimensional parameters which govern the critical current speed. These non-dimensional parameters, once computed, can then be used by riser designers to assess potential riser clashing and/or ascertain the minimum riser spacing and top tension required in order to avoid riser clashing.

Numerical and Experimental Investigations on Flexible Multi-bearing Rotor Dynamics

2004 ◽  
Vol 127 (4) ◽  
pp. 408-415 ◽  
Author(s):  
Q. Ding ◽  
A. Y. T. Leung
Keyword(s):  
Angular Acceleration ◽  
Rotor Dynamics ◽  
Dynamic Responses ◽  
Experimental Investigations ◽  
Test Rig ◽  
Theoretical Simulation ◽  
Speed Up ◽  
Simultaneous Existence ◽  
Stationary Bifurcation ◽  
The Mathematical Model

An experimental test rig is built to verify the dynamics of a multi-bearing rotor. It consists of two flexibly coupled shafts and is connected to a motor at one end via a flexible coupling. Each of the shafts is supported at the ends by two hydrodynamic bearings and is attached with two disks with equal and unequal masses, respectively. The mathematical model of the test rig is developed and is simulated numerically. The non-stationary dynamic responses of the system during speed-up with a constant angular acceleration are shown, respectively, by the non-stationary bifurcation diagrams, the selected time flows, and the spectrum cascades. Experiments are then carried out on the test rig. Generally, the numerical results are verified qualitatively by the experiments. Both results indicate that the non-synchronous whirls of the two shafts influence each other when flexibly coupled together. In particular, a new phenomenon is found for the four-bearing rotor system: the pre-existing non-synchronous whirl/whip resulted from the instability of one shaft can activate the onset of oil instability of another shaft. In the theoretical simulation, this phenomenon represents the rapid increase of the non-synchronous whirl orbit, whereas in the experiment, it represents the simultaneous existence of two whirl/whip frequencies in the spectra.

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