1,800,168 research outputs found
Rate of Decay of Stable Periodic Solutions of Duffing Equations
In this paper, we consider the second-order equations of Duffing type. Bounds
for the derivative of the restoring force are given that ensure the existence
and uniqueness of a periodic solution. Furthermore, the stability of the unique
periodic solution is analyzed; the sharp rate of exponential decay is
determined for a solution that is near to the unique periodic solution.Comment: Key words: Periodic solution; Stability; Rate of deca
Stability analysis for pitchfork bifurcations of solitary waves in generalized nonlinear Schroedinger equations
Linear stability of both sign-definite (positive) and sign-indefinite
solitary waves near pitchfork bifurcations is analyzed for the generalized
nonlinear Schroedinger equations with arbitrary forms of nonlinearity and
external potentials in arbitrary spatial dimensions. Bifurcations of
linear-stability eigenvalues associated with pitchfork bifurcations are
analytically calculated. It is shown that the smooth solution branch switches
stability at the bifurcation point. In addition, the two bifurcated solution
branches and the smooth branch have the opposite (same) stability when their
power slopes have the same (opposite) sign. One unusual feature on the
stability of these pitchfork bifurcations is that the smooth and bifurcated
solution branches can be both stable or both unstable, which contrasts such
bifurcations in finite-dimensional dynamical systems where the smooth and
bifurcated branches generally have opposite stability. For the special case of
positive solitary waves, stronger and more explicit stability results are also
obtained. It is shown that for positive solitary waves, their linear stability
near a bifurcation point can be read off directly from their power diagram.
Lastly, various numerical examples are presented, and the numerical results
confirm the analytical predictions both qualitatively and quantitatively.Comment: 28 pages, 6 figure
Analytical and numerical stability analysis of Soret-driven convection in a horizontal porous layer
We present an analytical and numerical stability analysis of Soret-driven convection in a porous cavity saturated by a binary fluid. Both the mechanical equilibrium solution and the monocellular flow obtained for particular ranges of the physical parameters of the problem are considered. The porous cavity, bounded by horizontal infinite or finite boundaries, is heated from below or from above. The two horizontal plates are maintained at different constant temperatures while no mass flux is imposed. The influence of the governing parameters and more particularly the role of the separation ratio, characterizing the Soret effect and the normalized porosity, are investigated theoretically and numerically. From the linear stability analysis, we find that the equilibrium solution loses its stability via a stationary bifurcation or a Hopf bifurcation depending on the separation ratio and the normalized porosity of the medium. The role of the porosity is important, when it decreases, the stability of the equilibrium solution is reinforced. For a cell heated from below, the equilibrium solution loses its stability via a stationary bifurcation when the separation ratio >0(Le,), while for 0, while a stationary or an oscillatory bifurcation occurs if mono the monocellular flow loses stability via a Hopf bifurcation. As the Rayleigh number increases, the resulting oscillatory solution evolves to a stationary multicellular flow. For a cell heated from above and <0, the monocellular flow remains linearly stable. We verified numerically that this problem admits other stable multicellular stationary solutions for this range of parameters
Stable self-similar blowup in energy supercritical Yang-Mills theory
We consider the Cauchy problem for an energy supercritical nonlinear wave
equation that arises in --dimensional Yang--Mills theory. A certain
self--similar solution of this model is conjectured to act as an
attractor for generic large data evolutions. Assuming mode stability of ,
we prove a weak version of this conjecture, namely that the self--similar
solution is (nonlinearly) stable. Phrased differently, we prove that mode
stability of implies its nonlinear stability. The fact that this
statement is not vacuous follows from careful numerical work by Bizo\'n and
Chmaj that verifies the mode stability of beyond reasonable doubt
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