Synchronization of a periodically forced Duffing oscillator with a periodically excited pendulum

In this paper, the analytical conditions for a periodically forced Duffing oscillator synchronized with a chaotic pendulum are developed through the theory of discontinuous dynamical systems. From the analytical conditions, the synchronization invariant domains are developed. For a better understanding of synchronization of two different dynamical systems, the partial and full synchronizations of the Duffing oscillator with the chaotic pendulum are presented for illustrations. The control parameter map is developed from the analytical conditions. Under special parameters, the two systems can be fully and partially synchronized. Since the forced pendulum has librational and rotational chaotic motions, the periodically forced Duffing oscillator can be synchronized only with the librational chaotic motions of the pendulum. It is impossible for the forced Duffing oscillator to be synchronized with the rotational chaotic motions.     

Non-linear flexings in a periodically forced floating beam

This paper considers periodic flexing in a floating beam, in the presence of a small periodic forcing term. The beam is considered as a vibrating beam with the free-end boundary condition, in the presence of an additional restoring force due to flotation, which becomes zero as soon as the beam lifts out of the water. The equation is therefore non-linear. A theorem is proved which shows that in the presence of small periodic forcing terms, both small- and large-amplitude solutions can exist. Numerical evidence is presented, which shows that the large-amplitude solutions are stable over a wide range of frequency and amplitude, and suggests a cusp-like surface for the multiple solutions.     

Multiple bifurcation trees of period-1 motions to chaos in a periodically forced,time-delayed,hardening Duffing oscillator

In this paper, bifurcation trees of periodic motions in a periodically forced, time-delayed, hardening Duffing oscillator are analytically predicted by a semi-analytical method. Such a semi-analytical method is based on the differential equation discretization of the time-delayed, nonlinear dynamical system. Bifurcation trees for the stable and unstable solutions of periodic motions to chaos in such a time-delayed, Duffing oscillator are achieved analytically. From the finite discrete Fourier series, harmonic frequency-amplitude curves for stable and unstable solutions of period-1 to period-4 motions are developed for a better understanding of quantity levels, singularity and catastrophes of harmonic amplitudes in the frequency domain. From the analytical prediction, numerical results of periodic motions in the time-delayed, hardening Duffing oscillator are completed. Through the numerical illustrations, the complexity and asymmetry of period-1 motions to chaos in nonlinear dynamical systems are strongly dependent on the distributions and quantity levels of harmonic amplitudes. With the quantity level increases of specific harmonic amplitudes, effects of the corresponding harmonics on the periodic motions become strong, and the certain complexity and asymmetry of periodic motion and chaos can be identified through harmonic amplitudes with higher quantity levels.     

A periodically forced,piecewise linear system. Part I: Local singularity and grazing bifurcation

A methodology for the local singularity of non-smooth dynamical systems is systematically presented in this paper, and a periodically forced, piecewise linear system is investigated as a sample problem to demonstrate the methodology. The sliding dynamics along the separation boundary are investigated through the differential inclusion theory. For this sample problem, a perturbation method is introduced to determine the singularity of the sliding dynamics on the separation boundary. The criteria for grazing bifurcation are presented mathematically and numerically. The grazing flows are illustrated numerically. This methodology can be very easily applied to predict grazing motions in other non-smooth dynamical systems. The fragmentation of the strange attractors of chaotic motion will be presented in the second part of this work.     

Chaos in periodically forced discrete-time ecosystem models

Natural populations whose generations are non-overlapping can be modelled by difference equations that describe how the populations evolve in discrete time-steps. These ecosystem models are, in general, nonlinear and contain system parameters that relate to such properties as the intrinsic growth-rate of a species. Typically, the parameters are kept constant. In this study, in order to simulate cyclic effects due to changes in environmental conditions, periodic forcing is applied to system parameters in four specific models, comprising three well-known, single-species models due to May, Moran–Ricker, and Hassell, and also a Maynard Smith predator–prey model. It is found that, in each case, a system that has simple (e.g., periodic) behavior in its unforced state can take on extremely complicated behavior, including chaos, when periodic forcing is applied, dependent on the values of the forcing amplitudes and frequencies. For each model, the application of forcing is found to produce an effective increase in the parameter space over which the system can behave chaotically. Bifurcation diagrams are constructed with the forcing amplitude as the bifurcation parameter, and these are observed to display rich structure, including chaotic bands with periodic windows, pitch-fork and tangent bifurcations, and attractor crises.     

Closed form solution of a periodically forced logistic model

We give the exact closed form solution of the following ordinary differential equation:

Grazing phenomena in a periodically forced,friction-induced,linear oscillator

The criterion for grazing motions in a dry-friction oscillator is obtained from the local theory of non-smooth dynamical systems on the connectable and accessible domains. The generic mappings for such a dry-friction oscillator are also introduced. The sufficient and necessary conditions for grazing at the final states of mappings are expressed. The initial and final switching sets of grazing mapping, varying with system parameters, are illustrated for the grazing parametric characteristics. The initial and grazing, switching manifolds in the switching sets are defined through grazing mappings. Finally, numerical illustrations of grazing motions are very easily carried out with help of the analytical predictions. This paper provides a comprehensive investigation of grazing motions in the dry-friction oscillator for a better understanding of the grazing mechanism of such a discontinuous system. The investigation based on the local singularity theory is more intuitive and efficient than the discontinuous mapping techniques.     

Connecting orbits for a periodically forced singular planar Newtonian system

In this paper we are concerned with the study of the existence and multiplicity of connecting orbits for a singular planar Newtonian system ${\ddot{q} + V_q(t, q) = 0}$ with a periodic strong force V _q (t, q), an infinitely deep well of Gordon's type at one point and two stationary points at which a potential V (t, q) achieves a strict global maximum. To this end we minimize the corresponding actiön functional over the classes of functions in the Sobolev space ${W^{1, 2}_{\rm loc}(\mathbb{R}, \mathbb{R}^2)}$ that turn a given number of times around the singularity.     

Chaotic dynamics in periodically forced asymmetric ordinary differential equations

Using a topological approach, we prove the existence of infinitely many periodic solutions and the presence of chaotic dynamics for the periodically forced second order ODE u^″+bu⁺−au⁻=p(t). The choice of the equation is motivated by the studies about the Dancer-Fu?ik spectrum and the Lazer-McKenna suspension bridge model.     

Attenuant cycles in periodically forced discrete-time age-structured population models

In discrete-time age-structured population models, a periodic environment is not always deleterious. We show that it is possible to have the average of the age class populations over an attracting cycle (in a periodic environment) not less than the average of the carrying capacities (in a corresponding constant environment). In our age-structured model, a periodic environment does not increase the average total biomass (no resonance). However, a periodic environment is disadvantageous for a population whenever there is no synchrony between the number of age classes and the period of the environment. As in periodically forced models without age-structure, we show that periodically forced age-structured population models support multiple attractors with complicated structures.     

On harmonic resonance in forced nonlinear oscillators exhibiting a Hopf bifurcation

The influence of a periodic forcing on a nonlinear second-orderoscillator close to a Hopf bifurcation point is investigated.The forcing frequency is close to the frequency of the Hopfbifurcation, and the forcing amplitude is assumed to be small.Second-order integral averaging is applied to reduce the givensystem to a planar autonomous system. By a bifurcation and stabilityanalysis of this system, the behaviour of the forced oscillatoris determined. It turns out that two qualitatively differenttypes of behaviour can occur. Either the system has a uniqueattractor, or the system has two competing attractors givingrise to a hysteresis phenomenon, which is known from the Duffingequation. Bifurcation diagrams are presented, and explicit formulaefor the quantities determining the behaviour are given     

Global chaos in a periodically forced, linear system with a dead-zone restoring force

The Poincare mapping and the corresponding mapping sections for global motions in a linear system possessing a dead-zone restoring force are introduced through switching planes pertaining to two constraints. The global periodic motions based on the Poincare mapping are determined, and the eigenvalue analysis for the stability and bifurcation of periodic motion is carried out. Global chaos in such a system is investigated numerically from the unstable global periodic motions analytically determined. The bifurcation scenario with varying parameters is presented. The mapping structures of periodic and chaotic motions are discussed. The Poincare mapping sections for global chaos are given for illustration. The grazing phenomenon embedded in chaotic motion is observed in this investigation.     

Flow switchability and periodic motions in a periodically forced,discontinuous dynamical system

This paper presents the switchability of a flow from one domain into another one in the periodically forced, discontinuous dynamical system. The inclined line boundary in phase space is used for the dynamical system to switch. The normal vector field product for flow switching on the separation boundary is introduced. The passability condition of a flow to the separation boundary is achieved through such a normal vector field product, and the sliding and grazing conditions to the separation boundary are presented as well. Using mapping structures, periodic motions in such a discontinuous system are predicted analytically, and the corresponding local stability and bifurcation analysis are carried out. With the analytical conditions of grazing and sliding motions, the parameter maps of specific motions are developed. Illustrations of periodic and chaotic motions are given, and the normal vector fields are presented to show the analytical criteria. This investigation may help one better understand the sliding mode control. The methodology presented in this paper can be applied to discontinuous, nonlinear systems.     

A geometric approach to periodically forced dynamical systems in presence of a separatrix

We consider the periodic boundary value problem for the non-autonomous scalar second-order equation , with e(·) a continuous and T-periodic forcing term. Using a continuation theorem adapted from Capietto et al. (Trans. Amer. Math. Soc. 329 (1992) 41-72), we propose some new conditions for the existence of T-periodic solutions to the forced equation in terms of the dynamical properties of the trajectories of the associated autonomous equation . Special emphasis will be addressed to the study of the case in which the presence of an unbounded separatrix for the autonomous system in the phase-plane allows to obtain a priori bounds for the T-periodic solutions of the homotopic equation .     

Steady state bifurcation of a periodically excited system under delayed feedback controls

This paper investigates the steady state bifurcation of a periodically excited system subject to time-delayed feedback controls by the combined method of residue harmonic balance and polynomial homotopy continuation. Three kinds of delayed feedback controls are considered to examine the effects of different delayed feedback controls and delay time on the steady state response. By means of polynomial homotopy continuation, all the possible steady state solutions corresponding the third-order superharmonic and second-subharmonic responses are derived analytically, i.e. without numerical integration. It is found that the delayed feedback changes the bifurcating curves qualitatively and possibly eliminates the saddle-node bifurcation during resonant. The delayed position-velocity coupling and the delayed velocity feedback controls can destabilize the steady state responses. Coexisting periodic solutions, period-doubling bifurcation and even chaos are found in these control systems. The neighborhood of the periodic solutions is verified numerically in the phase portraits. The various effects of time delay on the steady state response are investigated. Many new phenomena are observed.     

Bifurcation of sliding periodic orbits in periodically forced discontinuous systems

This paper is devoted to the study of bifurcations of periodic sliding solutions for discontinuous systems from sliding periodic solutions of unperturbed discontinuous equations. An example of 3-dimensional discontinuous ordinary differential equations is given to illustrate the theory.     

On motions and switchability in a periodically forced,discontinuous system with a parabolic boundary

The analytical conditions for motion switchability on the switching boundary in a periodically forced, discontinuous system are developed through the G-function of the vector fields to the switching boundary. Periodic motions in such a discontinuous dynamical system are discussed by the use of mapping structures. Two periodic motions and the analytical conditions are presented for illustration. Further investigation should be carried out for a better understanding of the vanishing and stability of regular and chaotic motions.     

Sliding and transversal motions on an inclined boundary in a periodically forced discontinuous system

In this paper, sliding and transversal motions on the boundary in the periodically driven, discontinuous dynamical system is investigated. The simple inclined straight line boundary in phase space is considered as a control law for such a dynamical system to switch. The normal vector field for a flow switching on the separation boundary is adopted to develop the analytical conditions, and the corresponding transversality conditions of a flow to the boundary are obtained. The conditions of sliding and grazing flows to the separation boundary are presented as well. Using mapping structures, periodic motions of such a discontinuous system are predicted, and the corresponding local stability and bifurcation analysis of the periodic motion are carried out. Numerical illustrations of periodic motions with and without sliding on the boundary are given. The local stability analysis cannot provide the proper prediction of the sliding and grazing motions in discontinuous dynamical systems. Therefore, the normal vector fields of periodic flows are presented, and the normal vector fields on the switching boundary points give the analytical criteria for sliding and transversality of motions.     

Long-time behavior of almost periodically forced parabolic equations on the circle

We study the long-time behavior of the skew-product semiflow generated by scalar reaction-diffusion equation on the circle with almost periodic forcing:

$u_t=u_{xx}+f(t,u,u_x),t>0,x\in S^1=\mathbb{R}/2\pi \mathbb{Z},$

where $f(t,u,u_x)$ is uniformly almost-periodic in t. Almost periodic environmental forcing exhibits the external effects which are roughly but not exactly periodic.Contrary to the time-periodic cases (for which any ω-limit set Ω can be embedded into a periodically forced circle flow), we show that, for almost-periodic forcing, the problem that whether Ω can be embedded into an almost-periodically forced circle flow is strongly related to the dimension of the center space $V^c(\mathrm{\Omega })$ associated with Ω. On the one hand, if $\dim V^c(\mathrm{\Omega })\le 1$ then Ω is either spatially-inhomogeneous or spatially-homogeneous; and moreover, any spatially-inhomogeneous Ω can be embedded into an almost periodically-forced circle flow. On the other hand, when $\dim V^c(\mathrm{\Omega })>1$, it is shown that the above embedding property cannot hold anymore. These reveal that for such system there are essential differences between time periodic forcing and non-periodic forcing.