Maximal Lyapunov Exponent and Rotation Number for Stochastically Perturbed Co-dimension Two Bifurcations

1994 ◽  
Vol 169 (3) ◽  
pp. 349-372 ◽  
Author(s):  
N. Sri Namachchivaya ◽  
S. Talwar
Keyword(s):  
Lyapunov Exponent ◽  
Rotation Number ◽  
Maximal Lyapunov Exponent ◽  
Stochastically Perturbed

Hamiltonian Chaos in a Model Alveolus

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
F. S. Henry ◽  
F. E. Laine-Pearson ◽  
A. Tsuda
Keyword(s):  
Reynolds Number ◽  
Fixed Point ◽  
Lyapunov Exponent ◽  
Fine Particles ◽  
Hamiltonian Dynamics ◽  
Maximal Lyapunov Exponent ◽  
Poincaré Sections ◽  
Pulmonary Acinus ◽  
Hamiltonian Chaos ◽  
Poincare Sections

In the pulmonary acinus, the airflow Reynolds number is usually much less than unity and hence the flow might be expected to be reversible. However, this does not appear to be the case as a significant portion of the fine particles that reach the acinus remains there after exhalation. We believe that this irreversibility is at large a result of chaotic mixing in the alveoli of the acinar airways. To test this hypothesis, we solved numerically the equations for incompressible, pulsatile, flow in a rigid alveolated duct and tracked numerous fluid particles over many breathing cycles. The resulting Poincaré sections exhibit chains of islands on which particles travel. In the region between these chains of islands, some particles move chaotically. The presence of chaos is supported by the results of an estimate of the maximal Lyapunov exponent. It is shown that the streamfunction equation for this flow may be written in the form of a Hamiltonian system and that an expansion of this equation captures all the essential features of the Poincaré sections. Elements of Kolmogorov–Arnol’d–Moser theory, the Poincaré–Birkhoff fixed-point theorem, and associated Hamiltonian dynamics theory are then employed to confirm the existence of chaos in the flow in a rigid alveolated duct.

Dynamic stability of running: The effects of speed and leg amputations on the maximal Lyapunov exponent

2013 ◽  
Vol 23 (4) ◽  
pp. 043131 ◽  
Author(s):  
Nicole Look ◽  
Christopher J. Arellano ◽  
Alena M. Grabowski ◽  
William J. McDermott ◽  
Rodger Kram ◽  
...  
Keyword(s):  
Lyapunov Exponent ◽  
Dynamic Stability ◽  
Maximal Lyapunov Exponent

FINITE DIMENSIONAL STRUCTURE OF THE GPI DISCHARGE IN PATIENTS WITH PARKINSON'S DISEASE

2011 ◽  
Vol 21 (03) ◽  
pp. 175-186 ◽  
Author(s):  
DANIELA SABRINA ANDRES ◽  
DANIEL CERQUETTI ◽  
MARCELO MERELLO
Keyword(s):  
Parkinson’S Disease ◽  
Parkinson's Disease ◽  
Lyapunov Exponent ◽  
Stochastic Systems ◽  
Rating Scale ◽  
Severe Disease ◽  
Dimensional Structure ◽  
Maximal Lyapunov Exponent ◽  
Embedding Dimension ◽  
Low Dimensional

Stochastic systems are infinitely dimensional and deterministic systems are low dimensional, while real systems lie somewhere between these two limit cases. If the calculation of a low (finite) dimension is in fact possible, one could conclude that the system under study is not purely random. In the present work we calculate the maximal Lyapunov exponent from interspike intervals time series recorded from the internal segment of the Globus Pallidusfrom patients with Parkinson's disease. We show the convergence of the maximal Lyapunov exponent at a dimension equal to 7 or 8, which is therefore our estimation of the embedding dimension for the system. For dimensions below 7 the observed behavior is what would be expected from a stochastic system or a complex system projecting onto lower dimensional spaces. The maximal Lyapunov exponent did not show any differences between tremor and akineto-rigid forms of the disease. However, it did decay with the value of motor Unified Parkinson's Disease Rating Scale -OFF scores. Patients with a more severe disease (higher UPDRS-OFF score) showed a lower value of the maximal Lyapunov exponent. Taken together, both indexes (the maximal Lyapunov exponent and the embedding dimension) remark the importance of taking into consideration the system's non-linear properties for a better understanding of the information transmission in the basal ganglia.

Scaling-law for the maximal Lyapunov exponent

1992 ◽  
Vol 25 (18) ◽  
pp. 4813-4826 ◽  
Author(s):  
R Livi ◽  
A Politi ◽  
S Ruffo
Keyword(s):  
Lyapunov Exponent ◽  
Scaling Law ◽  
Maximal Lyapunov Exponent
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