1,721,035 research outputs found
THE STRETCHING ALONG THE PATHS: GENESIS, EVOLUTION AND APPLICATIONS
No full textA brief and very partial survey of the stretching along the paths method is given, outlining which were the initial ideas, where they originated from, and how they have been developed in the last twenty five years or so
THE STRETCHING ALONG THE PATHS: GENESIS, EVOLUTION AND APPLICATIONS
Full text linkA brief and very partial survey of the stretching along the paths method is given, outlining which were the initial ideas, where they originated from, and how they have been developed in the last twenty five years or so
Global exponential stability of the periodic solution of a delayed neural network with discontinuous activations
No full textWe study the stability of a delayed Hopfield neural network with periodic coefficients and inputs and an arbitrary and constant delay. We consider non-decreasing activation functions which may also have jump discontinuities in order to model the ideal situation where the gain of the neuron amplifiers is very high and tends to infinity. In particular, we drop the assumption of Lipschitz continuity on the activation functions, which is usually required in most of the papers. Under suitable assumptions on the interconnection matrices, we prove that the delayed neural network has a unique periodic solution which is globally exponentially stable independently of the size of the delay. The assumptions we exploit concern the theory of M-matrices and are easy to check. Due to the possible discontinuities of the activation functions, the convergence of the output of the neural network is also studied by a suitable notion of limit. The existence, uniqueness and continuability of the solution of suitable initial value problems are proved
Infinitely many solutions for a Floquet-type BVP with superlinearity indefinite in sign
No full textBy the application of a technique developed by G. J. Butler we find infinitely many solutions of a Floquet-type BVP for the equation x''+q(t)g(x)=0, where q is a weight function that is allowed to change sign, g is is superlinear and such that g(x)x>0 for all non-zero x. The boundary condition is (x(b),x'(b))=L(x(a),x'(a)), where L is a continuous, positively homogeneous, and nondegenerate map. At first we apply the main result to obtain solutions with a prescribed large number of zeros when L is the rotation of a fixed angle l; second, we find infinitely many subharmonic solutions of any order and, again, solutions with a prescribed large number of zeros for the periodic problem associated to the equation x''+cx'+q(t)g(x)=0, with q and g as above and a constant c
Unbounded Solutions to Systems of Differential Equations at Resonance
Full text linkWe deal with a weakly coupled system of ODEs of the type xj′′+nj2xj+hj(x1,...,xd)=pj(t),j=1,...,d,with hj locally Lipschitz continuous and bounded, pj continuous and 2 π-periodic, nj∈ N (so that the system is at resonance). By means of a Lyapunov function approach for discrete dynamical systems, we prove the existence of unbounded solutions, when either global or asymptotic conditions on the coupling terms h1, ... , hd are assumed
Analysis of a lagoon ecological model with anoxic crises and impulsive harvesting
Full text linkWe analyze mathematically a system of impulsive nonlinear parabolic equations that model a shallow lagoon subject to anoxic crises and two types of impulsive harvesting. The main focus is on the existence and properties of periodic solutions. In particular we give conditions that ensure the existence of such solutions and examine the effect of harvesting on the occurrence of anoxic crises. Our approach is based on estimates of the principal eigenvalue of associated linear problems, and on results from Nonlinear Functional Analysis. In particular, we obtain explicit criteria that involve the integrals of coefficients rather than maxima and minima. This is significant due to the large seasonal variations in the coefficient values
Periodic solutions of asymptotically linear second order equations with changing sign weight
No full textIn this paper we study the ordinary differential equation x'' + q(t)g(x) = 0, where g is a locally Lipschitz continuous function that satisfies g(x)x > 0 for all non zero x and is asymptotically linear, while q is a continuous, π-periodic and changing sign weight. By the application of a recent result on the existence and multiplicity of fixed points of planar maps, we give conditions on q and on the behavior of the ratio g(x)/x near zero and near infinity in order to obtain multiple periodic solutions with the prescribed number of zeros in the intervals of positivity and negativity of q, as well as multiple subharmonics of any order and uncountably many bounded solutions
Periodic solutions to a forced kepler problem in the plane
Full text linkGiven a smooth function U(t, x), T-periodic in the first variable and satisfying U(t, x) = O(vertical bar x vertical bar(alpha)) for some alpha is an element of (0, 2) as vertical bar x vertical bar -> infinity, we prove that the forced Kepler problem(sic) = -x/vertical bar x vertical bar(3) + del U-x(t, x), x is an element of R-2,has a generalized T-periodic solution, according to the definition given in the paper by A. Boscaggin, R. Ortega, and L. Zhao [Trans. Amer. Math. Soc. 372 (2019), 677-703]. The proof relies on variational arguments
Nearly-circular periodic solutions of perturbed relativistic Kepler problems: the fixed-period and the fixed-energy problems
Full text linkThe paper studies the existence of periodic solutions of a perturbed relativistic Kepler problem of the type (Formula presented.) with d=2 or d=3, bifurcating, for ε small enough, from the set of circular solutions of the unperturbed system. Both the case of the fixed-period problem (assuming that U is T-periodic in time) and the case of the fixed-energy problem (assuming that U is independent of time) are considered
Chaotic dynamics in a periodically perturbed Liénard system
No full textWe prove the existence of infinitely many periodic solutions, as well as the presence of chaotic dynamics, for a periodically perturbed planar Liénard system of the form x' = y−F(x)+p(ωt), y' = −g(x). We consider the case in which the perturbing term is not necessarily small. Such a result is achieved by a topological method, that is by proving the presence of a horseshoe structure
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