A systematic approach to nonresonance conditions for periodically forced planar Hamiltonian systems

In the first part of the paper we consider periodic perturbations of some planar Hamiltonian systems. In a general setting, we detect conditions ensuring the existence and multiplicity of periodic solutions. In the second part, the same ideas are used to deal with some more general planar differential systems.

Influence of the Rotor-Driven Perturbation on the Stator-Exit Flow within a High-Pressure Gas Turbine Stage

In stator–rotor interaction studies on axial turbines, the attention is commonly focused on the unsteady rotor aerodynamics resulting from the periodic perturbations induced by the stator flow structures. Conversely, less interest has been historically attracted regarding the influence of the rotor on the flow released by the stator, correlated to propagation of the blade potential field upstream of the rotor leading edge. In this paper, experiments in the research high-pressure turbine of the Laboratory of Fluid-Machines of the Politecnico di Milano, performed by applying a fast-response aerodynamic pressure probe, alongside fully-3D time-accurate CFD simulations of the flow, are combined with the aim of discussing the rotor-to-stator interaction. While rotating, the rotor induces periodic perturbations on the pressure and velocity field in the stator–rotor gap, altering the evolution of the total quantities and the flow rate discharged by each stator channel and eventually triggering energy-separation effects which result in total pressure and total temperature oscillations in the stator-exit flow. Such oscillations were found to rise up to almost ±10% of the stage total temperature drop.

On resonances under quasi-periodic perturbations of systems with a double limit cycle, close to two-dimensional nonlinear Hamiltonian systems

The effect of multi-frequency quasi-periodic perturbations on systems close to twodimensional nonlinear Hamiltonian ones is studied. It is assumed that the corresponding perturbed autonomous system has a double limit cycle. Analysis of the Poincar´e–Pontryagin function constructed for the autonomous system makes it possible to establish the presence of such a cycle. When the condition of commensurability of the natural frequency of the corresponding unperturbed Hamiltonian system with the frequencies of the quasi-periodic perturbation is fulfilled, the unperturbed level becomes resonant. Resonant structures essentially depend on whether the selected resonance levels coincide with the levels that generate limit cycles in the autonomous system. An averaged system is obtained that describes the topology of the neighborhoods of resonance levels. Possible phase portraits of the averaged system are established near the bifurcation case, when the resonance level coincides with the level in whose neighborhood the corresponding autonomous system has a double limit cycle. To illustrate the results obtained, the results of a theoretical study and of a numerical calculation are presented for a specific pendulum-type equation under two-frequency quasi-periodic perturbations.

Tipping points induced by parameter drift in an excitable low-order model of the wind-driven ocean circulation

<p>A quasi-geostrophic, low-order model of the wind-driven ocean circulation is used to illustrate tipping points induced by time-dependent forcing in excitable chaotic systems. When the wind stress amplitude G is constant in time, our model has a bifurcation from low-amplitude oscillations to high-amplitude relaxation oscillations (ROs) at a wind intensity value G<sub>c</sub>. In the presence of time-dependent wind stress, the corresponding tipping point time t<sub>tp</sub> is defined as the time at which ROs arise. Numerical experiments are carried out using ensemble simulations in the presence of different drift rates of monotonically increasing forcing. Additional experiments include small periodic perturbations of such forcing. The results indicate substantial sensitivity of t<sub>tp</sub> and G(t<sub>tp</sub>) Rate-induced tipping, coexisting pullback attractors and total independence from initial states are found for subsets of parameter space. Besides, nonlinear resonance occurs in the presence of periodic perturbations for periods comparable to the RO time scale. The small periodic perturbation can be thought of as the seasonal-to-interannual variability in the wind stress, while the monotonically increasing component stands for the effect of amplification in the midlatitude winds due to anthropogenic warming.</p>

On two-frequency quasi-periodic perturbations of systems close to two-dimensional Hamiltonian ones with a double limit cycle

The problem of the effect of two-frequency quasi-periodic perturbations on systems close to arbitrary nonlinear two-dimensional Hamiltonian ones is studied in the case when the corresponding perturbed autonomous systems have a double limit cycle. Its solution is important both for the theory of synchronization of nonlinear oscillations and for the theory of bifurcations of dynamical systems. In the case of commensurability of the natural frequency of the unperturbed system with frequencies of quasi-periodic perturbation, resonance occurs. Averaged systems are derived that make it possible to ascertain the structure of the resonance zone, that is, to describe the behavior of solutions in the neighborhood of individual resonance levels. The study of these systems allows determining possible bifurcations arising when the resonance level deviates from the level of the unperturbed system, which generates a double limit cycle in a perturbed autonomous system. The theoretical results obtained are applied in the study of a two-frequency quasi-periodic perturbed pendulum-type equation and are illustrated by numerical computations.

Periodic Perturbations of a Class of Functional Differential Equations

We study the existence of a connected “branch” of periodic solutions of T-periodic perturbations of a particular class of functional differential equations on differentiable manifolds. Our result is obtained by a combination of degree-theoretic methods and a technique that allows to associate the bounded solutions of the functional equation to bounded solutions of a suitable ordinary differential equation.