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Multiple disjointness and invariant measures on minimal distal flows

Juho Rautio (2015)

Studia Mathematica

We examine multiple disjointness of minimal flows, that is, we find conditions under which the product of a collection of minimal flows is itself minimal. Our main theorem states that, for a collection X i i ∈ I of minimal flows with a common phase group, assuming each flow satisfies certain structural and algebraic conditions, the product ∏ i ∈ I X i is minimal if and only if ∏ i ∈ I X i e q is minimal, where X i e q is the maximal equicontinuous factor of X i . Most importantly, this result holds when each X i is distal. When the phase...

Multiple existence and stability of steady-states for a prey-predator system with cross-diffusion

Kousuke Kuto, Yoshio Yamada (2004)

Banach Center Publications

This article discusses a prey-predator system with cross-diffusion. We obtain multiple positive steady-state solutions of this system. More precisely, we prove that the set of positive steady-states possibly contains an S or ⊃-shaped branch with respect to a bifurcation parameter in the large cross-diffusion case. Next we give some criteria on the stability of these positive steady-states. Furthermore, we find the Hopf bifurcation point on the steady-state solution branch in a certain case. Our...

Multiple solutions of indefinite elliptic systems via a Galerkin-type Conley index theory

Marek Izydorek, Krzysztof P. Rybakowski (2003)

Fundamenta Mathematicae

Let Ω be a bounded domain in ℝ N with smooth boundary. Consider the following elliptic system: - Δ u = ∂ v H ( u , v , x ) in Ω, - Δ v = ∂ u H ( u , v , x ) in Ω, u = 0, v = 0 in ∂Ω. (ES) We assume that H is an even "-"-type Hamiltonian function whose first order partial derivatives satisfy appropriate growth conditions. We show that if (0,0) is a hyperbolic solution of (ES), then (ES) has at least 2|μ| nontrivial solutions, where μ = μ(0,0) is the renormalized Morse index of (0,0). This proves a conjecture by Angenent and van der Vorst.

Multiplicative integrable models from Poisson-Nijenhuis structures

Francesco Bonechi (2015)

Banach Center Publications

We discuss the role of Poisson-Nijenhuis (PN) geometry in the definition of multiplicative integrable models on symplectic groupoids. These are integrable models that are compatible with the groupoid structure in such a way that the set of contour levels of the hamiltonians in involution inherits a topological groupoid structure. We show that every maximal rank PN structure defines such a model. We consider the examples defined on compact hermitian symmetric spaces studied by F. Bonechi, J. Qiu...

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