Interaction of periodic and stationary bifurcation from multiple eigenvalues

1986 ◽  
Vol 192 (1) ◽  
pp. 159-166 ◽  
Author(s):  
Hansj�rg Kielh�fer
Keyword(s):  
Multiple Eigenvalues ◽  
Stationary Bifurcation

scholarly journals Iterative Methods for the Computation of the Perron Vector of Adjacency Matrices

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1522
Author(s):  
Anna Concas ◽  
Lothar Reichel ◽  
Giuseppe Rodriguez ◽  
Yunzi Zhang
Keyword(s):  
Iterative Methods ◽  
Adjacency Matrix ◽  
Lanczos Method ◽  
Power Method ◽  
Arnoldi Method ◽  
Multiple Eigenvalues ◽  
Computer Storage ◽  
Adjacency Matrices ◽  
Lanczos Methods ◽  
Perron Vector

The power method is commonly applied to compute the Perron vector of large adjacency matrices. Blondel et al. [SIAM Rev. 46, 2004] investigated its performance when the adjacency matrix has multiple eigenvalues of the same magnitude. It is well known that the Lanczos method typically requires fewer iterations than the power method to determine eigenvectors with the desired accuracy. However, the Lanczos method demands more computer storage, which may make it impractical to apply to very large problems. The present paper adapts the analysis by Blondel et al. to the Lanczos and restarted Lanczos methods. The restarted methods are found to yield fast convergence and to require less computer storage than the Lanczos method. Computed examples illustrate the theory presented. Applications of the Arnoldi method are also discussed.

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.

scholarly journals Axial Piston Pump Fault Diagnosis Method Based on Symmetrical Polar Coordinate Image and Fuzzy C-Means Clustering Algorithm

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Wan-lu Jiang ◽  
Pei-yao Zhang ◽  
Man Li ◽  
Shu-qing Zhang
Keyword(s):  
Fault Diagnosis ◽  
Clustering Algorithm ◽  
Piston Pump ◽  
Axial Piston Pump ◽  
Fuzzy C Means ◽  
Multiple Eigenvalues ◽  
Polar Coordinate ◽  
Fuzzy C Means Clustering ◽  
Diagnosis Method ◽  
Axial Piston

In this paper, a fault diagnosis method based on symmetric polar coordinate image and Fuzzy C-Means clustering algorithm is proposed to solve the problem that the fault signal of axial piston pump is not intuitive under the time-domain waveform diagram. In this paper, the sampled vibration signals of axial piston pump were denoised firstly by the combination of ensemble empirical mode decomposition and Pearson correlation coefficient. Secondly, the data, after noise reduction, was converted into images, called snowflake images, according to symmetric polar coordinate method. Different fault types of axial piston pump can be identified by observing the snowflake images. After that, in order to evaluate the research results objectively, the obtained images were converted into Gray-Level Cooccurrence Matrixes. Their multiple eigenvalues were extracted, and the eigenvectors consisting of multiple eigenvalues were classified by Fuzzy C-Means clustering algorithm. Finally, according to the accuracy of classification results, the feasibility of applying the symmetric polar coordinate method to axial piston pump fault diagnosis has been validated.

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