Table of contents

Consider a family of planar systems depending on two parameters (n, b) and having at most one limit cycle. Assume that the limit cycle disappears at some homoclinic (or heteroclinic) connection when Φ(n, b) = 0. We present a method that allows us to obtain a sequence of explicit algebraic lower and upper bounds for the bifurcation set Φ(n, b) = 0. The method is applied to two quadratic families, one of them is the well-known Bogdanov–Takens system. One of the results that we obtain for this system is the bifurcation curve for small values of n, given by . We obtain the new three terms from purely algebraic calculations, without evaluating Melnikov functions.

Some general results on the energy stability/instability of droplets with zero contact angle for thin film type equations including those governed by the power laws are established. Energy instability of droplets with nonzero contact angle and configurations of droplets is also obtained. As an application, an asymptotic stability result for droplets with zero contact angle is established.

We study numerically the long-time evolution of the surface quasi-geostrophic equation with generalized viscosity of the form (−▵)^α, where global regularity has been proved mathematically for the subcritical parameter range α ⩾ 1/2. Even in the supercritical range, we have found numerically that smooth evolution persists, but with a very slow and oscillatory damping in the long run. A subtle balance between nonlinear and dissipative terms is observed therein. Notably, qualitative behaviours of the analytic properties of the solution do not change in the super and subcritical ranges, suggesting the current theoretical boundary α = 1/2 is of technical nature.

Suppose that the real generic cubic Hamiltonian H(x, y), , possesses three saddle points and one centre. Let be the set of values h of H(x, y), for which there exists a closed component δ(h) of the level curve {H(x, y) = h}, free of critical points. In this paper, we obtain a better upper bound than previously known for the number of zeros of the Abelian integrals I(h) = ∫_δ(h)[g(x, y) dx − f(x, y) dy] for h ∊ Σ in terms of the maximum of the degrees of the polynomials f(x, y) and g(x, y).

We prove the existence of a partially strong solution to the steady Navier–Stokes equations for barotropic compressible fluid in a bounded simply connected domain with the prescribed generalized impermeability conditions u · n = 0, curl u · n = 0 and curl²u · n = 0 on the boundary. We assume that the state law for the pressure has the form for γ > 3. We call the solution 'partially strong' because only the divergence-free part of velocity and the effective pressure have regularity typical for strong solutions, while the gradient part of velocity and the density have regularity typical for weak solutions.

Mode-locking regions (resonance tongues) formed by border-collision bifurcations of piecewise-smooth, continuous maps commonly exhibit a distinctive sausage-like geometry with pinch points called 'shrinking points'. In this paper we extend our unfolding of the piecewise-linear case [Simpson and Meiss (2009 Nonlinearity22 1123–44)] to show how shrinking points are destroyed by nonlinearity. We obtain a codimension-three unfolding of this shrinking point bifurcation for N-dimensional maps. We show that the destruction of the shrinking points generically occurs by the creation of a curve of saddle-node bifurcations that smooth one boundary of the sausage, leaving a kink in the other boundary.

We consider the one-dimensional initial value problem for the viscous transport equation with nonlocal velocity u_t = u_xx − (u(K' * u))_x with a given kernel . We show the existence of global-in-time nonnegative solutions and we study their large time asymptotics. Depending on K', we obtain either linear diffusion waves (i.e. the fundamental solution of the heat equation) or nonlinear diffusion waves (the fundamental solution of the viscous Burgers equation) in asymptotic expansions of solutions as t → ∞. Moreover, for certain aggregation kernels, we show a concentration of solution on an initial time interval, which resemble a phenomenon of the spike creation, typical in chemotaxis models.

We present a planar agent-based flocking model with a distance-dependent communication weight. We derive a sufficient condition for the asymptotic flocking in terms of the initial spatial and heading-angle diameters and a communication weight. For this, we employ differential inequalities for the spatial and phase diameters together with the Lyapunov functional approach. When the diameter of the agent's initial heading-angles is sufficiently small, we show that the diameter of the heading-angles converges to the average value of the initial heading-angles exponentially fast. As an application of flocking estimates, we also show that the Kuramoto model with a connected communication topology on the regular lattice for identical oscillators exhibits a complete-phase-frequency synchronization, when coupled oscillators are initially distributed on the half circle.

Isotropic–nematic phase transition due to excluded volume effects is considered for the 2D Onsager model of rigid rod-like polymers with a general interaction kernel involving a finite number of Fourier modes. Nematic phases appear above a concentration threshold. We prove the existence of continuous branches of nematic states bifurcating from the isotropic phase at concentrations proportional to the Fourier coefficients of the interaction kernel. The bifurcation structure is shown to depend on the size of the spectral gaps of the interaction operator. Exact multiplicity of nematic phases is proved for a class of two-mode trigonometric kernels. Our arguments use simple bifurcation and variational tools applied to an equivalent finite dimensional problem that involves multi-variable Bessel functions.

We show that if a singularity of suitable weak solutions to Navier–Stokes equations occurs, then either p or at least two of ∂_iv_i, i = 1, 2, 3, have neither upper bounds nor lower bounds in any neighbourhood of the singularity. In the case of axially symmetric solutions, we prove that either p or ∂_rv^r is not bounded both below and above near a singular point, if it exists.

Using the general theory of Hopf bifurcation with symmetry we study here the example where the group of symmetries is O(3), the rotations and reflections of a sphere. We make some amendments to previously published lists of C-axial isotropy subgroups of O(3) × S¹ and list the isotropy subgroups with four-dimensional fixed-point subspaces. We then study the particular example where O(3) × S¹ acts on the space V₃ ⊕ V₃ where V₃ is the space of spherical harmonics of degree three. We find that in this case there are six C-axial isotropy subgroups of O(3) × S¹. The equivariant Hopf theorem guarantees the existence of periodic solutions with each of these symmetries in O(3) × S¹ equivariant differential equations. Three of the solutions are found to be standing waves and the other three are travelling waves. We compute conditions for each of these solution branches to be stable and by restricting the O(3) × S¹ equivariant differential equations to four-dimensional invariant subspaces we are able to find additional periodic and quasiperiodic solutions.

We prove two results about hyperbolic periodic solutions in networks of systems of ODEs. First, we show that generically hyperbolic periodic solutions of network admissible systems of differential equations oscillate in each node if and only if the network is transitive. We can associate a polydiagonal Δ(Z(t)) with each hyperbolic periodic solution Z(t) as follows. The cell coordinates of a point in Δ(Z(t)) are equal if the corresponding cell coordinates of Z(t) are equal for all t; that is, the outputs from the two cells are synchronous. Second, we prove that Δ(Z(t)) is rigid (unchanged by small admissible perturbations) if and only if it is flow-invariant for all admissible vector fields.

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