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We study global continuation properties of the set of T-periodic solutions of parameterized second order delay differential equations with constant time lag on smooth manifolds. We apply our results to get multiplicity of T-periodic solutions. Our topological approach is mainly based on the notion of degree of a tangent vector field.

We provide some new estimates for the topological degree of a class of continuous and piecewise linear maps based on a classical integral computation formula. We provide applications to some nonlinear problems that exhibit a local P C 1 structure. [ABSTRACT FROM AUTHOR]

The relation between heat flux and temperature gradient has been considered as a constitutive structure or as a balance law in different approaches. Both views may allow a description of heat conduction characterized by finite speed propagation of temperature disturbances. Such a result, which overcomes Fourier's drawback of infinite speed propagation, can be obtained also by considering insufficient the representation of a conductor, even when it is considered to be rigid, rather than the sole relation between heat flux and temperature gradient. We comment this last view and describe the intersection with previous proposals. Eventually, we show how under Fourier's law we can have traveling-wave-type temperature propagation when thermal microstructures are accounted for. [ABSTRACT FROM AUTHOR]

This paper provides sufficient conditions for any map L, that is strongly piecewise linear relatively to a decomposition of Rk in admissible cones, to be invertible. Namely, via a degree theory argument, we show that when there are at most four convex pieces (or three pieces with at most a non convex one), the map is invertible. Examples show that the result cannot be plainly extended to a greater number of pieces. Our result is obtained by studying the structure of strongly piecewise linear maps. We then extend the results to the PC¹ case. [ABSTRACT FROM AUTHOR]

We find multiplicity results for forced oscillations of a periodically perturbed autonomous second‐order equation, the perturbing term possibly depending on the whole history of the system. The techniques that we use are topological in nature, but the technical details are hidden in the proofs and completely transparent to the reader only interested in the results. [ABSTRACT FROM AUTHOR]

We study the set of T-periodic solutions of a class of T-periodically perturbed coupled and nonautonomous differential equations on manifolds. By using degree-theoretic methods we obtain a global continuation result for the T-periodic solutions of the considered equations. [ABSTRACT FROM AUTHOR]

In this paper we deal with the strong local optimality of a triplet satisfying Pontryagin Maximum Principle in the minimum time problem between two given manifolds. The reference control is assumed to be bang-bang with a double switching time. Our methods are based on a topological technique for the inversion of the projected maximised flow. [ABSTRACT FROM AUTHOR]

By using topological methods, mainly the degree of a tangent vector field, we establishmultiplicity results for T-periodic solutions of parametrized T-periodic perturbations of autonomous ODEs on a differentiable manifold M. In order to provide insights into the key notion of T-resonance, we consider the elementary situations M=R and M=R². Doing so, we provide more comprehensive analysis of those cases and find improved conditions. [ABSTRACT FROM AUTHOR]

We study forced oscillations on differentiable manifolds which are globally defined as the zero set of appropriate smooth maps in some Euclidean spaces. Given a T-periodic perturbative forcing field, we consider the two different scenarios of a nontrivial unperturbed force field and of perturbation of the zero field. We provide simple, degree-theoretic conditions for the existence of branches of T-periodic solutions. We apply our construction to a class of second-order differential-algebraic equations. [ABSTRACT FROM AUTHOR]

This paper gives sufficient conditions for a class of bang-bang extremals with multiple switches to be locally optimal in the strong topology. The conditions are the natural generalizations of those considered in [A. A. Agrachev, G. Stefani, and P. Zezza, SI AM J. Control Optim., 41 (2002), pp. 991-1014], [L. Poggiolini, Rend. Semin. Mat. Univ. Politec. Torino, 64 (2006), pp. 1-23], and [L. Poggiolini and G. Stefani, Systems Control Lett, 53 (2004), pp. 269-279]. We require both the strict bang-bang Legendre condition and the second order conditions for the finite dimensional problem obtained by moving the switching times of the reference trajectory. [ABSTRACT FROM AUTHOR]

We study the set of harmonic solutions to perturbed periodic separated variables ordinary differential equations on manifolds. As an application, a multiplicity result is deduced. [ABSTRACT FROM AUTHOR]

Using a topological approach we investigate the structure of the set of forced oscillations of a class of reducible second order functional retarded differential equations subject to periodic forcing. More precisely, we consider a delay-type functional dependence involving a gamma probability distribution and, using a linear chain trick, we formulate a first order system of ODEs whose T-periodic solutions correspond to those of the functional equation. [ABSTRACT FROM AUTHOR]

We prove a multiplicity result for forced oscillations of a spherical pendulum (that is, a massive point moving on a sphere) subject to a periodic action, with or without friction, allowed to depend on the whole past of the motion. The approach is based on topological methods. In particular, when the unperturbed forcing term is the gravity, we obtain two harmonic forced oscillations regardless of the presence of friction and of the form of the perturbing force field. [ABSTRACT FROM AUTHOR]

• Explicit traveling-wave-type solutions to the Maxwell-Cattaneo scheme. • Asymptotic analysis of a nonlinear modification of Maxwell-Cattaneos scheme. • Foundations of Capriz-Wilmanski-Marianos non-linear non-Fourier heat conduction. We compare traveling-wave-type solutions to Maxwell-Cattaneo's description of heat transfer with those of Capriz-Wilmanski-Mariano's version, which includes nonlinearities due to microstructural contributions in the presence of heat-flux-driven phase transition in a conductor otherwise rigid. Our closed-form results for the nonlinear scheme, based on asymptotic-type analysis, show how temperature traveling-wave-type evolution is perturbed by the influence of microstructural effects. Also, our results underline how even in Maxwell-Cattaneo's scheme initial conditions in terms of heat flux and temperature have some hidden link, although in the scheme these variables are a priori independent with each other. [ABSTRACT FROM AUTHOR]