ESCAPING DYNAMICS IN THE PRESENCE OF DISSIPATION AND NOISE IN SCATTERING SYSTEMS

2010 ◽  
Vol 20 (09) ◽  
pp. 2783-2793 ◽  
Author(s):  
JESÚS M. SEOANE ◽  
MIGUEL A. F. SANJUÁN
Keyword(s):  
Fractal Dimension ◽  
Decay Rate ◽  
Exponential Decay ◽  
Scaling Law ◽  
Scattering Function ◽  
Chaotic Scattering ◽  
Fundamental Interest ◽  
External Perturbations ◽  
Exponential Decay Rate ◽  
Scattering Systems

Chaotic scattering in open Hamiltonian systems is a problem of fundamental interest with applications in several branches of physics. In this paper we analyze the effects of adding external perturbations such as dissipation and noise in chaotic scattering phenomena. Our main result is the exponential decay rate of the particles in the scattering region when the system is affected by dissipation and noise. In the case of dissipation the particles escape more slowly from the scattering region than in the conservative case. However, in the noisy case, the particles escape faster from the scattering region as compared to the noiseless case. Moreover, we analyze the fractal dimension of the set of singularities of the scattering function for the dissipative and the conservative cases. As a result of our analysis we have found that a scaling law exists between the exponential decay rate of the particles and the dissipative parameter, and that the fractal dimension for the noisy case is the unity.

scholarly journals Assessment of Patellar Tendon Reflex Responses Using Second-Order System Characteristics

2016 ◽  
Vol 2016 ◽  
pp. 1-4 ◽  
Author(s):  
Brett D. Steineman ◽  
Pavan Karra ◽  
Kiwon Park
Keyword(s):  
Decay Rate ◽  
Natural Frequency ◽  
Patellar Tendon ◽  
Exponential Decay ◽  
Second Order ◽  
Order System ◽  
Sample Population ◽  
Exponential Decay Rate ◽  
Tendon Reflex ◽  
Patellar Tendon Reflex

Deep tendon reflex tests, such as the patellar tendon reflex (PTR), are widely accepted as simple examinations for detecting neurological disorders. Despite common acceptance, the grading scales remain subjective, creating an opportunity for quantitative measures to improve the reliability and efficacy of these tests. Previous studies have demonstrated the usefulness of quantified measurement variables; however, little work has been done to correlate experimental data with theoretical models using entire PTR responses. In the present study, it is hypothesized that PTR responses may be described by the exponential decay rate and damped natural frequency of a theoretical second-order system. Kinematic data was recorded from both knees of 45 subjects using a motion capture system and correlation analysis found that the meanR2value was 0.99. Exponential decay rate and damped natural frequency ranges determined from the sample population were −5.61 to −1.42 and 11.73 rad/s to 14.96 rad/s, respectively. This study confirmed that PTR responses strongly correlate to a second-order system and that exponential decay rate and undamped natural frequency are novel measurement variables to accurately measure PTR responses. Therefore, further investigation of these measurement variables and their usefulness in grading PTR responses is warranted.

scholarly journals Feedback stabilization  of a 3D fluid flow by shape deformations of an obstacle

2021 ◽  
Author(s):  
Jean-Pierre Raymond ◽  
Muthusamy Vanninathan
Keyword(s):  
Fluid Flow ◽  
Decay Rate ◽  
Exponential Decay ◽  
Stokes Equations ◽  
Feedback Stabilization ◽  
Navier Stokes ◽  
Initial Shape ◽  
Exponential Decay Rate ◽  
Time Dependent Domain ◽  
Initial Location

We consider a fluid flow in a time dependent domain $\Omega_f(t)=\Omega \setminus \Omega_s(t)\subset {\mathbb R}^3$, surrounding a deformable obstacle $\Omega_s(t)$. We assume that the fluid flow satisfies the incompressible Navier-Stokes equations in  $\Omega_f(t)$, $t>0$. We prove that, for any arbitrary exponential decay rate $\omega>0$, if the initial condition of the fluid flow is small enough in some norm, the deformation of the boundary $\partial \Omega_s(t)$ can be chosen so that  the fluid flow is  stabilized to rest, and the obstacle to its initial shape and its initial location, with the  exponential decay rate $\omega>0$.

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