scholarly journals Dispersion of discontinuous periodic waves

2013 ◽  
Vol 469 (2149) ◽  
pp. 20120407 ◽  
Author(s):  
Gong Chen ◽  
Peter J. Olver
Keyword(s):  
Dispersion Relation ◽  
Numerical Experiments ◽  
Polynomial Growth ◽  
Dynamic Evolution ◽  
Bounded Domains ◽  
Periodic Waves ◽  
Talbot Effect ◽  
Asymptotically Linear ◽  
Dispersive Waves ◽  
Wave Phenomena

The dynamic evolution of linearly dispersive waves on periodic domains with discontinuous initial profiles is shown to depend remarkedly upon the asymptotics of the dispersion relation at large wavenumbers. Asymptotically linear or sublinear dispersion relations produce slowly changing waves, while those with polynomial growth exhibit dispersive quantization, a.k.a. the Talbot effect, being (approximately) quantized at rational times, but a non-differentiable fractal at irrational times. Numerical experiments suggest that such effects persist into the nonlinear regime, for both integrable and non-integrable systems. Implications for the successful modelling of wave phenomena on bounded domains and numerical challenges are discussed.

scholarly journals Are Numerical Simulations Reliable? Proposed Improvements and Tests

1987 ◽  
Vol 117 ◽  
pp. 279-279
Author(s):  
F. R. Bouchet
Keyword(s):  
Growth Rate ◽  
Dispersion Relation ◽  
Numerical Experiments ◽  
Linear Growth ◽  
Numerical Dispersion ◽  
Linear Growth Rate ◽  
Linear Regime ◽  
Grid Spacing ◽  
Gravitational Clustering ◽  
Theoretical Predictions

When one builds a code to simulate numerically a process, the first concern is the range of validity of the results. This can be accessed empirically, though the results can be misleading if the tests are too naive. For particle-mesh codes simulating the gravitational clustering, an analytical theory has been proposed in Bouchet et al. 1985. It yields the numerical dispersion relation of the system in the linear regime, and thus describes how the linear growth rate is affected by the discretisation. The theoretical predictions are in agreement with the results of actual numerical experiments: both show that the results of standart particle-mesh codes should not be trusted at distances smaller than 6 to 8 grid-spacing Δx (depending on the detail of the algorithm).

scholarly journals Analysis of Stability-To-Chaos in the Dynamic Evolution of Network Traffic Flows under a Dual Updating Mechanism

2021 ◽  
Author(s):  
Shixu Liu ◽  
Hao Yan ◽  
Said M. Easa ◽  
Lidan Guo ◽  
Yingnuo Tang
Keyword(s):  
Traffic Flow ◽  
Network Traffic ◽  
Numerical Experiments ◽  
Dynamic Theory ◽  
Evolutionary Model ◽  
Dynamic Evolution ◽  
Traffic Flows ◽  
Route Network ◽  
Analysis Of Stability ◽  
Route Cost

This paper proposes a traffic-flow evolutionary model under a dual updating mechanism that describes the day-to-day (DTD) dynamics of traffic flow and travel cost. To illustrate the concept, a simple two-route network is considered. Based on the nonlinear dynamic theory, the equilibrium stability condition of the system is derived and the condition for the division between the bifurcation and chaotic states of the system is determined. The characteristics of the DTD dynamic evolution of network traffic flow are investigated using numerical experiments. The results show that the system is absolutely stable when the sensitivity of travelers toward the route cost parameter (θ) is equal to or less than 0.923. The bifurcation appears in the system when θ is larger than 0.923. For values of θ equal to or larger than 4.402, the chaos appears in the evolution of the system. The results also show that with the appearance of chaos, the boundary and interior crises begin to appear in the system when θ is larger than 6.773 and 10.403, respectively. The evolution of network traffic flow is always stable when the proportion of travelers who do not change the route is 84% or greater.

Asymptotic behaviour of Markov population processes by asymptotically linear rate of change

1994 ◽  
Vol 31 (3) ◽  
pp. 614-625 ◽  
Author(s):  
F. C. Klebaner
Keyword(s):  
Markov Processes ◽  
Growth Rates ◽  
Continuous Time ◽  
Polynomial Growth ◽  
Rate Of Change ◽  
Asymptotically Linear ◽  
Linear Rate ◽  
Type Distribution ◽  
Linear Differential ◽  
Markov Population Processes

Multidimensional Markov processes in continuous time with asymptotically linear mean change per unit of time are studied as randomly perturbed linear differential equations. Conditions for exponential and polynomial growth rates with stable type distribution are given. From these conditions results on branching models of populations with stabilizing reproduction for near-supercritical and near-critical cases follow.

scholarly journals Dispersive waves in functionally graded plates

2018 ◽  
Vol 251 ◽  
pp. 04052
Author(s):  
Alla Ilyashenko ◽  
Sergey Kuznetsov
Keyword(s):  
Dispersion Relation ◽  
Functionally Graded ◽  
Harmonic Waves ◽  
Functionally Graded Plates ◽  
Dispersive Waves ◽  
Fg Plates

The dispersion waves propagating in anisotropic functionally graded (FG) plates with arbitrary transverse heterogeneity and arbitrary elastic monoclinic anisotropy are analysed within a recently developed six-dimensional formalism. The dispersion relation is obtained for all modes of dispersive harmonic waves propagating in an unbounded plate.

scholarly journals Analysis of Stability-To-Chaos in the Dynamic Evolution of Network Traffic Flows under a Dual Updating Mechanism

2021 ◽  
Author(s):  
Shixu Liu ◽  
Hao Yan ◽  
Said M. Easa ◽  
Lidan Guo ◽  
Yingnuo Tang
Keyword(s):  
Traffic Flow ◽  
Network Traffic ◽  
Numerical Experiments ◽  
Dynamic Theory ◽  
Evolutionary Model ◽  
Dynamic Evolution ◽  
Traffic Flows ◽  
Route Network ◽  
Analysis Of Stability ◽  
Route Cost

This paper proposes a traffic-flow evolutionary model under a dual updating mechanism that describes the day-to-day (DTD) dynamics of traffic flow and travel cost. To illustrate the concept, a simple two-route network is considered. Based on the nonlinear dynamic theory, the equilibrium stability condition of the system is derived and the condition for the division between the bifurcation and chaotic states of the system is determined. The characteristics of the DTD dynamic evolution of network traffic flow are investigated using numerical experiments. The results show that the system is absolutely stable when the sensitivity of travelers toward the route cost parameter (θ) is equal to or less than 0.923. The bifurcation appears in the system when θ is larger than 0.923. For values of θ equal to or larger than 4.402, the chaos appears in the evolution of the system. The results also show that with the appearance of chaos, the boundary and interior crises begin to appear in the system when θ is larger than 6.773 and 10.403, respectively. The evolution of network traffic flow is always stable when the proportion of travelers who do not change the route is 84% or greater.

scholarly journals ON THE INTERACTION OF ELASTIC WAVES

2003 ◽  
Vol 9 (3) ◽  
pp. 218-224 ◽  
Author(s):  
Aleksandras Krylovas ◽  
Raimondas Čiegis
Keyword(s):  
Mathematical Model ◽  
Asymptotic Expansions ◽  
Elastic Waves ◽  
Numerical Experiments ◽  
Resonant Interaction ◽  
Periodic Waves ◽  
Non Linear

The non-linear mathematical model of the interaction of elastic waves is presented. The conditions of possible resonant interaction of periodic waves are described. The method of internal averaging for getting uniformly valid asymptotic expansions is used in both, ie resonant and non-resonant, cases. Results of numerical experiments are presented for the resonant interaction of the elastic waves.

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