We establish the existence of a time-periodic carrying simplex for competitive systems of Kolmogorov-type ordinary differential equations of the form ẋ_i = x_i g_i(t,x) , i=1,… ,d , on ℝ^d_+ , where the vector field g is T-periodic in time and satisfies only Carathéodory conditions rather than the classical C^1 regularity assumed in earlier work. This relaxed regularity requires a careful reworking of global existence, uniqueness, and continuous dependence results for the associated initial value problem on the closed first orthant. We show that the time-T Poincaré map P generated by the flow is retrotone, and apply recent results on the carrying simplex for retrotone maps to obtain for P a carrying simplex Σ that is an unordered, Lipschitz, invariant manifold homeomorphic to the standard probability simplex, and that attracts all nonzero trajectories of the discrete dynamical system generated by P. We then extend this construction to the full nonautonomous flow by introducing an auxiliary autonomous system on the extended phase space 𝒯×ℝ^d_+ , where 𝒯 is the circle of length T, generated via a skew-product formulation. In the context of this extended phase space we then construct a compact global attractor and identify, as its relative boundary, a time-periodic family of carrying simplices {Σ _s}_s ∈𝒯 that depend continuously on the phase s, each homeomorphic to the standard simplex and each attracting, uniformly in s, all nonzero orbits of the corresponding time-shifted flow. We further show that the restriction of the extended flow to the time-periodic carrying simplex is topologically conjugate, via orthogonal projection, to a flow generated by a reduced time-periodic Carathéodory system on a (d-1) -dimensional linear subspace, extending prior results in the autonomous setting to the nonautonomous, low-regularity case. These results provide a rigorous foundation for the qualitative analysis of seasonally forced competitive population models in which growth rates need only be measurable and locally Lipschitz in time, broadening the applicability of carrying simplex theory to more realistic ecological settings.
This paper investigates the well-posedness of a class of one-dimensional Boussinesq-type systems modeling nonlinear dispersive water waves over varying bottom topographies and offers the first rigorous analytical framework for the initial-value formulation of the Madsen and Sørensen Boussinesq-type equations. The equations under consideration depend on a real parameter B, which modulates the strength of the dispersive effect, and on the static water depth h_0 . In the case of flat bathymetry, we provide a detailed analysis of the associated key operator, formulate the quasilinear system, and construct a suitable symmetrizer to define an appropriate energy norm. Local well-posedness in the "hyperbolic" space is established under three distinct assumptions on the pair (B, h_0) . The analysis is subsequently extended to the variable bathymetry setting in the critical case B = -1/6 , which is the only value for which the model admits a mathematically tractable structure allowing for a rigorous treatment. For this setting, we derive a symmetrization and prove local well-posedness results under natural conditions.
In this paper we study the dynamics on the invariant spheres for two classes of 3-dimensional polynomial differential systems of the form ẋ=x+Q(x), x=(x,y,z)∈ℝ^3. First, when Q(x) is linear we characterize these differential systems having invariant the unit sphere centered at the origin of coordinates and describe their phase portraits depending on the parameters of the system. Second, when Q(x) is a cubic homogeneous polynomial we classify these differential systems having invariant the unit sphere centered at the origin of coordinates and we describe the dynamics on this sphere depending on the parameters of the system.
This paper establishes the theoretical result concerning the stability of non-zero point of pullback random attractors in a regular space. To be precise, we prove the upper semicontinuity of pullback random attractors for non-autonomous random dynamical systems with delay parameter ρ as ρ tends to ρ _0 ( 0<ρ <ρ _0 ). Since the attractors of the original system and the corresponding limiting system belong to two different spaces, the main difficulty lies in how to place the attractors of these two systems in the same space. In the application part, we consider stochastic retarded magnetohydrodynamics (MHD) equations with non-autonomous forcing term. Since the solutions of MHD equations lack higher regularity for initial data belonging to a space of limited smoothness, we prove the pullback asymptotic compactness of solution operators for such equations via the spectrum decomposition technique, and then obtain the existence and uniqueness of regular pullback random attractors.
In this paper we consider a nonisotropic harmonic oscillator with Coriolis coupling, perturbed by a generalized cubic Hénon-Heiles potential. This model depends on three parameters which play an important role in the symmetry breaking of the classical Hénon-Heiles system. The main goal is to describe the equilibrium solutions of the system and their linear stability character. We also analyze the nonlinear stability of the equilibrium point at the origin, showing how the symmetry breaking, which manifests itself through the parameters of the problem, affects the stability properties through resonances.
