scholarly journals Synchronous Steady State Solutions of a Symmetric Mixed Cubic-Superlinear Schrödinger System

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 190
Author(s):  
Riadh Chteoui ◽  
Abdulrahman F. Aljohani ◽  
Anouar Ben Mabrouk
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Nehari Manifold ◽  
Nonlinear Pdes ◽  
Nonlinear Part ◽  
Numerical Investigations ◽  
Physical Phenomena ◽  
Steady State Solutions ◽  
Schrödinger System ◽  
Theoretical Results

Systems of coupled nonlinear PDEs are applied in many fields as suitable models for many natural and physical phenomena. This makes them active and attractive subjects for both theoretical and numerical investigations. In the present paper, a symmetric nonlinear Schrödinger (NLS) system is considered for the existence of the steady state solutions by applying a minimizing problem on some modified Nehari manifold. The nonlinear part is a mixture of cubic and superlinear nonlinearities and cubic correlations. Some numerical simulations are also illustrated graphically to confirm the theoretical results.

Swirling flow states in finite-length diverging or contracting circular pipes

2017 ◽  
Vol 819 ◽  
pp. 678-712 ◽  
Author(s):  
Zvi Rusak ◽  
Yuxin Zhang ◽  
Harry Lee ◽  
Shixiao Wang
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Finite Length ◽  
Vortex Breakdown ◽  
Steady States ◽  
Inviscid Limit ◽  
Circular Pipes ◽  
Outlet Flow ◽  
Separation Zones ◽  
Steady State Solutions

The dynamics of inviscid-limit, incompressible and axisymmetric swirling flows in finite-length, diverging or contracting, long circular pipes is studied through global analysis techniques and numerical simulations. The inlet flow is described by the profiles of the circumferential and axial velocity together with a fixed azimuthal vorticity while the outlet flow is characterized by a state with zero radial velocity. A mathematical model that is based on the Squire–Long equation (SLE) is formulated to identify steady-state solutions of the problem with special conditions to describe states with separation zones. The problem is then reduced to the columnar (axially-independent) SLE, with centreline and wall conditions for the solution of the outlet flow streamfunction. The solution of the columnar SLE problem gives rise to the existence of four types of solutions. The SLE problem is then solved numerically using a special procedure to capture states with vortex-breakdown or wall-separation zones. Numerical simulations based on the unsteady vorticity circulation equations are also conducted and show correlation between time-asymptotic states and steady states according to the SLE and the columnar SLE problems. The simulations also shed light on the stability of the various steady states. The uniqueness of steady-state solutions in a certain range of swirl is proven analytically and demonstrated numerically. The computed results provide the bifurcation diagrams of steady states in terms of the incoming swirl ratio and size of pipe divergence or contraction. Critical swirls for the first appearance of the various types of states are identified. The results show that pipe divergence promotes the appearance of vortex-breakdown states at lower levels of the incoming swirl while pipe contraction delays the appearance of vortex breakdown to higher levels of swirl and promotes the formation of wall-separation states.

Dynamics and pattern formation in a modified Leslie–Gower model with Allee effect and Bazykin functional response

2017 ◽  
Vol 10 (05) ◽  
pp. 1750073 ◽  
Author(s):  
Peng Feng
Keyword(s):  
Steady State ◽  
Pattern Formation ◽  
Spatial Pattern ◽  
Numerical Simulations ◽  
Functional Response ◽  
Allee Effect ◽  
Turing Instability ◽  
Spatial Pattern Formation ◽  
The Stability ◽  
Steady State Solutions

In this paper, we study the dynamics of a diffusive modified Leslie–Gower model with the multiplicative Allee effect and Bazykin functional response. We give detailed study on the stability of equilibria. Non-existence of non-constant positive steady state solutions are shown to identify the rage of parameters of spatial pattern formation. We also give the conditions of Turing instability and perform a series of numerical simulations and find that the model exhibits complex patterns.

Dynamic Analysis of the Optical Disk Drive System Equipped With an Automatic Ball Balancer With Consideration of Torsional Motion

2003 ◽  
Author(s):  
Chun-Chieh Wang ◽  
Cheng-Kuo Sung ◽  
Paul C. P. Chao
Keyword(s):  
Steady State ◽  
Multiple Scales ◽  
Substantial Reduction ◽  
Disk Drive ◽  
Optical Disk ◽  
Balance System ◽  
Optical Disk Drive ◽  
The Stability ◽  
Steady State Solutions ◽  
Theoretical Results

This study is dedicated to evaluate the stability of an automatic ball-type balance system (ABS) installed in Optical Disk Drives (ODD). There have been researchers devoted to study the performance of ABS by investigating the dynamics of the system, but few consider the motions in torsional direction of ODD foundation. To solve this problem, a mathematical model including the foundation is established. The method of multiple scales is then utilized to find all possible steady-state solutions and perform related stability analysis. The obtained results are used to predict the level of residual vibrations and then the performance of the ABS can be evaluated. Numerical simulations are conducted to verify the theoretical results. It is obtained from both analytical and numerical results that the spindle speed of the motor ought to be operated above primary translational and secondary torsional resonances to stabilize the desired steady-state solutions for a substantial reduction in radial vibration.

scholarly journals Bifurcations and Dynamics of the Rb-E2F Pathway Involving miR449

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-20
Author(s):  
Lingling Li ◽  
Jianwei Shen
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Theory ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Delay Model ◽  
Dynamical Behaviors ◽  
Theoretical Results

We focused on the gene regulative network involving Rb-E2F pathway and microRNAs (miR449) and studied the influence of time delay on the dynamical behaviors of Rb-E2F pathway by using Hopf bifurcation theory. It is shown that under certain assumptions the steady state of the delay model is asymptotically stable for all delay values; there is a critical value under another set of conditions; the steady state is stable when the time delay is less than the critical value, while the steady state is changed to be unstable when the time delay is greater than the critical value. Thus, Hopf bifurcation appears at the steady state when the delay passes through the critical value. Numerical simulations were presented to illustrate the theoretical results.

scholarly journals Modeling magnetized star-planet interactions: boundary conditions effects

2013 ◽  
Vol 8 (S300) ◽  
pp. 330-334 ◽  
Author(s):  
Antoine Strugarek ◽  
Allan Sacha Brun ◽  
Sean P. Matt ◽  
Victor Reville
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Numerical Simulations ◽  
Stellar Wind ◽  
Stellar Surface ◽  
Numerical Studies ◽  
Rotational Evolution ◽  
Further Development ◽  
Steady State Solutions

AbstractWe model the magnetized interaction between a star and a close-in planet (SPMIs), using global, magnetohydrodynamic numerical simulations. In this proceedings, we study the effects of the numerical boundary conditions at the stellar surface, where the stellar wind is driven, and in the planetary interior. We show that is it possible to design boundary conditions that are adequate to obtain physically realistic, steady-state solutions for cases with both magnetized and unmagnetized planets. This encourages further development of numerical studies, in order to better constrain and undersand SPMIs, as well as their effects on the star-planet rotational evolution.

STABILITY OF SPATIALLY PERIODIC SOLUTIONS IN COUPLED MAP LATTICES

2003 ◽  
Vol 13 (02) ◽  
pp. 343-356 ◽  
Author(s):  
M. DUBCOVÁ ◽  
A. KLÍČ ◽  
P. POKORNÝ ◽  
D. TURZÍK
Keyword(s):  
Steady State ◽  
Periodic Solutions ◽  
Nonlinear Evolution ◽  
Linear Operators ◽  
Coupled Map Lattices ◽  
Evolution Operators ◽  
Spatially Periodic ◽  
The Stability ◽  
Steady State Solutions ◽  
Theoretical Results

Stability of steady-state solutions of 1-dim coupled map lattices is studied. The stability is determined by the spectrum of linear operators on two-sided sequences of vectors in [Formula: see text] arising as a linearization of the corresponding nonlinear evolution operators. Theoretical results are applied to several examples.

On the Dynamics of Vibration Absorbers With Motion-Limiting Stops

1998 ◽  
Vol 65 (1) ◽  
pp. 223-233 ◽  
Author(s):  
H. Luo ◽  
S. Hanagud
Keyword(s):  
Steady State ◽  
Mechanical Model ◽  
Piecewise Linear ◽  
Nonlinear Effects ◽  
Vibration Absorbers ◽  
Chaotic Vibrations ◽  
Piecewise Linear System ◽  
Period 2 ◽  
Steady State Solutions ◽  
Theoretical Results

The dynamics of a class of vibration absorbers with elastic stops is discussed in this paper. The mechanical model proposed in previously published papers are modified to explain certain nonlinear effects, chaotic vibrations, and lower damping observed in our studies. Refined contact-noncontact criteria are presented. Exact steady-state solutions are obtained for a piecewise linear system by using the proposed contact-noncontact criteria. Numerical simulations are presented and compared with the results of the previous work. Significant differences that have been found include some chaotic responses of the system. Experiments are conducted to validate the theoretical results. Chaotic and period-2 responses are also detected experimentally.

Steady State Bifurcation and Patterns of Reaction–Diffusion Equations

2020 ◽  
Vol 30 (11) ◽  
pp. 2050215
Author(s):  
Chunrui Zhang ◽  
Baodong Zheng
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Spatial Region ◽  
Bifurcation Points ◽  
Steady State Bifurcation ◽  
Theoretical Results

In this paper, steady state bifurcations arising from the reaction–diffusion equations are investigated. Using the Lyapunov–Schmidt reduction on a square domain, a simple, and a double steady state bifurcation caused by the symmetry of spatial region is obtained. By examining the reduced bifurcation equations, complete bifurcation scenario and patterns at simple and double steady state bifurcation points are obtained. Numerical simulations support the theoretical results.

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