Systems with Phase Space Dimension N ≥ 3: Deterministic Chaos

2014 ◽  
pp. 75-92
Author(s):  
Vadim S. Anishchenko ◽  
Tatyana E. Vadivasova ◽  
Galina I. Strelkova
Keyword(s):  
Phase Space ◽  
Space Dimension ◽  
Deterministic Chaos

Chaos in blood flow control in genetic and renovascular hypertensive rats

1991 ◽  
Vol 261 (3) ◽  
pp. F400-F408 ◽  
Author(s):  
K. P. Yip ◽  
N. H. Holstein-Rathlou ◽  
D. J. Marsh
Keyword(s):  
Time Series ◽  
Blood Flow ◽  
Phase Space ◽  
Renal Artery ◽  
Initial Conditions ◽  
Pressure Fluctuations ◽  
Deterministic Chaos ◽  
Tubuloglomerular Feedback ◽  
Hypertensive Rats ◽  
Statistical Measures

Hydrostatic pressure and flow in renal proximal tubules oscillate at 30–40 mHz in normotensive rats anesthetized with halothane. The oscillations originate in tubuloglomerular feedback, a mechanism that provides local blood flow regulation. Instead of oscillations, spontaneously hypertensive rats (SHR) have aperiodic tubular pressure fluctuations; the pattern is suggestive of deterministic chaos. Normal rats made hypertensive by clipping one renal artery had similar aperiodic tubular pressure fluctuations in the unclipped kidney, and the fraction of rats with irregular fluctuations increased with time after the application of the renal artery clip. Statistical measures of deterministic chaos were applied to tubular pressure data. The correlation dimension, a measure of the dimension of the phase space attractor generating the time series, indicated the presence of a low-dimension strange attractor, and the largest Lyapunov exponent, a measure of the rate of divergence in phase space, was positive, indicating sensitivity to initial conditions. These time series therefore satisfy two criteria of deterministic chaos. The measures were the same in SHR as in rats with renovascular hypertension. Since two different models of hypertension displayed similar dynamics, we suggest that chaotic behavior is a common feature of renal vascular control in the natural history of the disease.

scholarly journals Chaotic Statistical Downscaling (CSD): Application and Comparison in the Bogotá River Basin

10.29007/wkcx ◽  
2018 ◽  
Author(s):  
Freddy Duarte ◽  
Gerald Corzo ◽  
Germán Santos ◽  
Oscar Hernández
Keyword(s):  
Time Delay ◽  
Phase Space ◽  
Predictive Model ◽  
River Basin ◽  
Statistical Downscaling ◽  
Climate Model ◽  
Global Climate ◽  
Global Climate Model ◽  
Deterministic Chaos ◽  
Climatic Information

This study presents a new statistical downscaling method called Chaotic Statistical Downscaling (CSD). The method is based on three main steps: Phase space reconstruction for different time steps, identification of deterministic chaos and a general synchronization predictive model. The Bogotá river basin was used to test the methodology. Two sources of climatic information are downscaled: the first corresponds to 47 rainfall gauges stations (1970-2016, daily) and the second is derived from the information of the global climate model MPI-ESM-MR with a resolution of 1,875° x 1,875° daily resolution. These time series were used to reconstruct the phase space using the Method of Time-Delay. The Time-Delay method allows us to find the appropriate values of the time delay and the embedding dimension to capture the dynamics of the attractor. This information was used to calculate the exponents of Lyapunov, which shows the existence of deterministic chaos. Subsequently, a predictive model is created based on the general synchronization of two dynamical systems. Finally, the results obtained are compared with other statistical downscaling models for the Bogota River basin using different measures of error which show that the proposed method is able to reproduce reliable rainfall values (RMSE=73.37).

TIME SERIES ANALYSIS OF COMPLEX DYNAMICAL BEHAVIOR CONTAMINATED WITH OBSERVATIONAL NOISE

1996 ◽  
Vol 06 (11) ◽  
pp. 2031-2045 ◽  
Author(s):  
TAKAYA MIYANO
Keyword(s):  
Time Series ◽  
Phase Space ◽  
Random Noise ◽  
Dynamical Behavior ◽  
Deterministic Chaos ◽  
Diagnostic Algorithm ◽  
Diagnostic Methods ◽  
Observational Noise ◽  
Nonlinear Forecasting ◽  
Complex Dynamical Behavior

Diagnostic methods for discovering deterministic chaos based on the instability and the parallelness of nearby trajectories generated from a time series in phase space are applied to numerical time series contaminated with additive random noise. The diagnostic algorithm based on nonlinear forecasting is prone to be fooled when handling chaotic data including observational noise. Such a misdiagnosis can be circumvented by estimating the degrees of parallelness of neighboring trajectories in the phase space. Dynamical properties of global temperature variations and voice signals of Japanese vowel /a/ are examined by the combinational use of the diagnostic algorithms.

CHAOTIC ANALYSES FOR SPACE SERIES OF GOLD GRADE

2004 ◽  
Vol 18 (17n19) ◽  
pp. 2730-2733 ◽  
Author(s):  
YAN-SHI XIE ◽  
GUANG-HAO CHEN ◽  
KAI-XUAN TAN
Keyword(s):  
Fractal Dimension ◽  
Phase Space ◽  
Lyapunov Exponent ◽  
Dynamic Process ◽  
Chaotic Attractor ◽  
Space Dimension ◽  
Chaotic Dynamic ◽  
Largest Lyapunov Exponent ◽  
Gold Grade ◽  
Chaotic Theory

A new powerful tool, chaotic theory, has been used to study mineralization through chaotic analysis for space series of gold grade in this paper. Both of the most important chaotic measures, Largest Lyapunov exponent (LLE) and fractal dimensional, for space series of gold grade in one gold deposit are computed. The positive LLE suggests that the space series of gold grade are chaotic series. When the phase space dimension approach 8~10, a chaotic attractor appears and their fractal dimension values vary from 1.94 to 3.99. It indicates that the evolution of ore-forming fluid and the enrichment and deposition of gold element are chaotic dynamic process.

An electronic technique for measuring phase space dimension from chaotic time series

1988 ◽  
Vol 131 (2) ◽  
pp. 85-90 ◽  
Author(s):  
A. Namajūnas ◽  
J. Požela ◽  
A. Tamaševičius
Keyword(s):  
Time Series ◽  
Phase Space ◽  
Space Dimension ◽  
Chaotic Time Series

A ROUTE TO CHAOS IN ELECTROMECHANICAL SYSTEMS: PHASE SPACE ATTRACTION BASIN SWITCHING

2009 ◽  
Vol 19 (07) ◽  
pp. 2363-2375 ◽  
Author(s):  
MARCO A. MONTAVA BELDA
Keyword(s):  
Phase Space ◽  
State Space ◽  
Nonlinear Model ◽  
Chaotic Dynamics ◽  
Deterministic Chaos ◽  
Time Varying ◽  
Attraction Basin ◽  
Periodic Input ◽  
Saddle Type ◽  
Route To Chaos

Certain systems present chaotic dynamics when subjected to a regular periodic input. In a study of a nonlinear model of an electromechanical transducer, its dynamic stability is analyzed and it is observed to present chaotic dynamics when a squared signal is introduced as input to the excitor circuit voltage. It is demonstrated that the chaotic movement is due to the periodic modification in the attraction basin of the state space, caused by the input varying in time. Varying the input causes the system to cross saddle type bifurcation values in which points of equilibrium appear and disappear, periodically modifying the qualitative aspects of the system's phase space. This paper describes the deterministic chaos generation by the regular and periodic modification of the properties of the phase space.

PUPILLARY HIPPUS: AN UNRECOGNIZED EXAMPLE OF BIOLOGIC CHAOS

1999 ◽  
Vol 07 (01) ◽  
pp. 85-94 ◽  
Author(s):  
MICHAEL L. ROSENBERG ◽  
MARTIN H. KROLL
Keyword(s):  
Phase Space ◽  
Lyapunov Exponent ◽  
Pupil Size ◽  
Correlation Dimension ◽  
Deterministic Chaos ◽  
Period Doubling ◽  
Linear Process ◽  
Data Sets ◽  
Space Analysis ◽  
Disease States

We analyzed the pupil size vs time from six subjects using pupilography and nonlinear techniques. The correlation dimensions ranged from 4.08 to 5.7. The Hurst exponents ranged from 0.132 to 0.546. All data sets contained at least one positive Lyapunov exponent. The use of surrogate yielded statistically significant differences for the correlation dimension. Phase space analysis yields a definite flow, and in subject two, period-doubling is evident. The accumulated evidence supports the notion that dynamics of pupil size are governed by deterministic chaos rather than a stochastic or linear process. This implies that one might discern between well and disease states using pupillography and that the dynamics can be mechanistically modeled.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

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