Abstract
A periodic boundary value problem is considered for one of the first versions of the Kuramoto–Sivashinsky equation, which is widely known in mathematical physics. Local bifurcations in a neighborhood of spatially homogeneous equilibrium points are studied in the case when they change stability. It is shown that the loss of stability of homogeneous equilibrium points leads to the occurrence of a two-dimensional local attractor on which all solutions are periodic functions of time, except for one spatially inhomogeneous state. The spectrum of frequencies of this family of periodic solutions fills the entire number line, and all of them are unstable in the sense of Lyapunov’s definition in the metric of the phase space (the space of initial conditions) of the corresponding initial boundary value problem. As the phase space, a Sobolev functional space natural for this boundary value problem is chosen. Asymptotic formulas are given for periodic solutions filling the two-dimensional attractor. To analyze the bifurcation problem, analysis methods for an infinite-dimensional dynamical system are used: the integral (invariant) manifold method combined with the methods of the Poincare normal form theory and asymptotic methods. Analyzing the bifurcations for the periodic boundary value problem is reduced to analyzing the structure of the neighborhood of the zero solution to the homogeneous Dirichlet boundary value problem for the equation under consideration.
Highlights
Действительно, рассмотрим два различных решения из семейства периодических решений (11), т.е. up(t, x, c1, φ1, ε) и up(t, x, c2, φ2, ε) (c1 = c2) и выделим "главные" части в асимптотическом представлении для этих двух решений up(t, x, c1, φ1, ε) = wp(t, x, c1, φ1, ε) + o(ε), up(t, x, c2, φ2, ε) = wp(t, x, c2, φ2, ε) + o(ε), w1(t, x) = wp(t, x, c1, φ1, ε) = c1 + ε1/2 sin(x + σ1t + φ1), σ1 = −2c1, w2(t, x) = wp(t, x, c2, φ2, ε) = c2 + ε1/2 sin(x + σ2t + φ2), σ2 = −2c2
A Local Attractor Filled with Unstable Periodic Solutions", Modeling and Analysis of Information Systems, 25:1 (2018), 92–101
Summary
Во-вторых, если u(t, x) – ее решение, то π. ЛДО A(c), определенный на достаточно гладких функциях p(x), удовлетворяющих условиям (6), является производящим оператором аналитической полугруппы линейных ограниченных операторов в гильбертовом пространстве H0 : f (x) ∈ H0, если. Через Hk будем обозначать гильбертово пространство, состоящее из тех 2π периодических функций f (x), у которых существуют обобщенные производные до порядка k включительно, принадлежащие L2(−π, π). Что КЗ (5), (6) имеет нулевое состояние равновесия. В частности, в работе будут рассмотрены вопросы о поведении решений при t → ∞ вспомогательной КЗ (5), (6) с начальными условиями f (x) ∈ Q(r) ⊂ H4,0. Здесь через Q(r) обозначен шар радиуса r с центром в нуле фазового пространства решений КЗ (5), (6), т.е. Здесь через Q(r) обозначен шар радиуса r с центром в нуле фазового пространства решений КЗ (5), (6), т.е. шар с центром в нуле гильбертова пространства H4,0
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
