Bifurcation and stability of radially symmetric equilibria of a parabolic equation with variable diffusion

Nascimento, Arnaldo Simal do

doi:10.1016/0022-0396(89)90158-7

Cited by 9 publications

(4 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The shooting method used to construct the attractor generalizes the bifurcation result in [9] for radially symmetric solutions in the disk. Indeed, not only we are able to prove the existence of bifurcating equilibria, but can also compute secondary bifurcations that might occur, hyperbolicity of all equilibria, their Morse indices and how they fit together in the attractor, by computing heteroclinic trajectories.…”

Section: Discussionmentioning

confidence: 99%

“…A new pair of equilibria appears when λ crosses an eigenvalue of the spherical laplacian λ k . This characterizes the pitchfork bifurcations that occur at each λ k and gives a different proof of such results, as in [9]. Then there are 2k + 3 intersections of M u ∩ M s , and the angle of the tangent vector of the unstable shooting curve at (0, 0) ∈ M u π/2 is given by µ(λ k+1 ) = π 2 (k + 1).…”

Section: Intersection Of Shooting Curves: Finding Equilibriamentioning

confidence: 99%

Section: Construction Of the Shooting Curvesmentioning

confidence: 99%

See 2 more Smart Citations

Sturm Attractors for Quasilinear Parabolic Equations with Singular Coefficients

Lappicy

2018

J Dyn Diff Equat

View full text Add to dashboard Cite

The goal of this paper is to construct explicitly the global attractors of parabolic equations with singular diffusion coefficients on the boundary, as it was done without the singular term for the semilinear case by Brunovský and Fiedler (1986), generalized by Fiedler and Rocha (1996) and later for quasilinear equations by the author (2017). In particular, we construct heteroclinic connections between hyperbolic equilibria, stating necessary and sufficient conditions for heteroclinics to occur. Such conditions can be computed through a permutation of the equilibria. Lastly, an example is computed yielding the well known Chafee-Infante attractor.Keywords: parabolic equations, singular coefficients, infinite dimensional dynamical systems, global attractor, Sturm attractor. 0 < ǫ ≤ a(x, u, p) ≤ δ 1. the global attractor A of (1.2) consists of finitely many equilibria E and their heteroclinic orbits H.2. there exists a heteroclinic u(t) ∈ H between u − , u + ∈ E such thatif, and only if, u − and u + are adjacent and i(u − ) > i(u + ).The first claim follows due to the existence of a Lyapunov functional constructed by Matano [29] and Zelenyak [44]. A modification of such functional for the case of singular coefficients is done in Section 2.1.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Intersection Of Shooting Curves: Finding Equilibriamentioning

confidence: 99%

See 1 more Smart Citation

Sturm Attractors for Quasilinear Parabolic Equations with Singular Coefficients

Lappicy

2018

J Dyn Diff Equat

View full text Add to dashboard Cite

show abstract

“…For the case in which 0=B(1, 0), the unit n-dimensional ball, this author showed in [4], for k 2 #1, that if r=&X& and the following hypothesis is satisfied r 2 k 1, rr +(n&1) rk 1, r (n&1) k 1 (r), 0<r<1, then no stable nonconstant equilibrium solution of (1.2) exists.…”

Section: Introductionmentioning

confidence: 99%