Global Analysis of Nonlinear Time-Delayed Dynamical Systems

2011 ◽  
Author(s):  
Jian-Qiao Sun ◽  
Bo Song ◽  
Jie Yang
Keyword(s):  
Steady State ◽  
Dynamical Systems ◽  
State Space ◽  
Global Analysis ◽  
Stable Steady State ◽  
Domains Of Attraction ◽  
Infinite Dimensional ◽  
Dimensional State Space ◽  
Steady State Responses ◽  
State Responses

Time-delayed dynamical systems are defined in an infinite dimensional state-space. When the system has multiple stable steady-state responses, the global analysis of the system such as finding the domains of attraction and boundary basin has to be done in the infinite dimensional state-space. This paper examines this issue and shows that there are rich opportunities for global analysis research of nonlinear time-delayed dynamical systems.

Parallel Cell Mapping Method for Global Analysis of High-Dimensional Nonlinear Dynamical Systems1

2015 ◽  
Vol 82 (11) ◽  
Author(s):  
Fu-Rui Xiong ◽  
Zhi-Chang Qin ◽  
Qian Ding ◽  
Carlos Hernández ◽  
Jesús Fernandez ◽  
...  
Keyword(s):  
Steady State ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Global Analysis ◽  
Mapping Method ◽  
Cell Mapping ◽  
Nonlinear Dynamical ◽  
Cell Mapping Methods ◽  
Steady State Responses ◽  
State Responses

The cell mapping methods were originated by Hsu in 1980s for global analysis of nonlinear dynamical systems that can have multiple steady-state responses including equilibrium states, periodic motions, and chaotic attractors. The cell mapping methods have been applied to deterministic, stochastic, and fuzzy dynamical systems. Two important extensions of the cell mapping method have been developed to improve the accuracy of the solutions obtained in the cell state space: the interpolated cell mapping (ICM) and the set-oriented method with subdivision technique. For a long time, the cell mapping methods have been applied to dynamical systems with low dimension until now. With the advent of cheap dynamic memory and massively parallel computing technologies, such as the graphical processing units (GPUs), global analysis of moderate- to high-dimensional nonlinear dynamical systems becomes feasible. This paper presents a parallel cell mapping method for global analysis of nonlinear dynamical systems. The simple cell mapping (SCM) and generalized cell mapping (GCM) are implemented in a hybrid manner. The solution process starts with a coarse cell partition to obtain a covering set of the steady-state responses, followed by the subdivision technique to enhance the accuracy of the steady-state responses. When the cells are small enough, no further subdivision is necessary. We propose to treat the solutions obtained by the cell mapping method on a sufficiently fine grid as a database, which provides a basis for the ICM to generate the pointwise approximation of the solutions without additional numerical integrations of differential equations. A modified global analysis of nonlinear systems with transient states is developed by taking advantage of parallel computing without subdivision. To validate the parallelized cell mapping techniques and to demonstrate the effectiveness of the proposed method, a low-dimensional dynamical system governed by implicit mappings is first presented, followed by the global analysis of a three-dimensional plasma model and a six-dimensional Lorenz system. For the six-dimensional example, an error analysis of the ICM is conducted with the Hausdorff distance as a metric.

Estimating the Audiogram Using Multiple Auditory Steady-State Responses

2002 ◽  
Vol 13 (04) ◽  
pp. 205-224 ◽  
Author(s):  
Andrew Dimitrijevic ◽  
Sasha M. John ◽  
Patricia Van Roon ◽  
David W. Purcell ◽  
Julija Adamonis ◽  
...  
Keyword(s):  
Steady State ◽  
Correlation Coefficients ◽  
Behavioral Threshold ◽  
Behavioral Thresholds ◽  
Sensorineural Hearing Impairment ◽  
Steady State Responses ◽  
Auditory Steady State Responses ◽  
Tonal Stimuli ◽  
Conductive Hearing ◽  
State Responses

Multiple auditory steady-state responses were evoked by eight tonal stimuli (four per ear), with each stimulus simultaneously modulated in both amplitude and frequency. The modulation frequencies varied from 80 to 95 Hz and the carrier frequencies were 500, 1000, 2000, and 4000 Hz. For air conduction, the differences between physiologic thresholds for these mixed-modulation (MM) stimuli and behavioral thresholds for pure tones in 31 adult subjects with a sensorineural hearing impairment and 14 adult subjects with normal hearing were 14 ± 11, 5 ± 9, 5 ± 9, and 9 ± 10 dB (correlation coefficients .85, .94, .95, and .95) for the 500-, 1000-, 2000-, and 4000-Hz carrier frequencies, respectively. Similar results were obtained in subjects with simulated conductive hearing losses. Responses to stimuli presented through a forehead bone conductor showed physiologic-behavioral threshold differences of 22 ± 8, 14 ± 5, 5 ± 8, and 5 ± 10 dB for the 500-, 1000-, 2000-, and 4000-Hz carrier frequencies, respectively. These responses were attenuated by white noise presented concurrently through the bone conductor.

Weighted averaging of steady-state responses

2001 ◽  
Vol 112 (3) ◽  
pp. 555-562 ◽  
Author(s):  
M.Sasha John ◽  
Andrew Dimitrijevic ◽  
Terence W Picton
Keyword(s):  
Steady State ◽  
Weighted Averaging ◽  
Steady State Responses ◽  
State Responses
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