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In this paper, a novel morphing airfoil based on a four-corner simply supported bistable composite laminated shell is elaborated. Based on the thin airfoil theory, the aerodynamic load can be identified. Utilizing the Hamilton’s principle, dynamic equations concerning the dynamic snap-through and nonlinear dynamics of the novel morphing airfoil are elucidated. The inertia, damping, restoring force and aerodynamic load terms are accounted for in the dynamic equations. A visual representation of the restoring force curve is exhibited, revealing the snap-through and the three equilibrium states. A number of quasi-static locations are exhibited in the context of the restoring force curve, generating a number of related airfoils. Thus, a graphic representation of the aerodynamic coefficients at the pre-snap-through, snap-through and post-snap-through stages is provided. The novel morphing airfoil under minor disturbances is held constant, and its aerodynamically controlled motions are characterized by the period doubling bifurcation and the 1/2-subharmonic resonance, which are both depicted in a variety of phase and frequency spectrum diagrams. Shock dynamics under transient aerodynamic excitation for large disturbances are detailed, illuminating the limit cycle oscillations and the chaotic snap-through. The limit cycle oscillations, the multiple-period snap-through and the chaotic snap-through are features of the morphing airfoil under harmonic excitation. The negative stiffness system offered by the bistable composite laminated shell has a significant influence on the amplitude–frequency response curves’ apparent softening nonlinear effect. The development of aircraft morphing assemblies can be supported by this study.

This paper explores the chaotic dynamics of a piezoelectrically laminated initially curved microbeam resonator subjected to fringing-field electrostatic actuation, for the first time. The resonator is fully clamped at both ends and is coated with two piezoelectric layers, encompassing both the top and bottom surfaces. The nonlinear motion equation which is obtained by considering the nonlinear fringing-field electrostatic force, includes geometric nonlinearities due to the mid-plane stretching and initial curvature. The motion equation is discretized using Galerkin method and the reduced order system is numerically integrated over the time for the time response. The variation of the first three natural frequencies with respect to the applied electrostatic voltage is determined and the frequency response curve is obtained using the combination of shooting and continuation methods. The bifurcation points have been examined and their types have been clarified based on the loci of the Floquet exponents on the complex plane. The period-doubled branches of the frequency response curves originating from the period doubling (PD) bifurcation points are stablished. It's demonstrated that the succession PD cascades leads to chaotic behavior. The chaotic behavior is identified qualitatively by constructing the corresponding Poincaré section and analyzing the response's associated frequency components. The bifurcation diagram is obtained for a wide range of excitation frequency and thus the exact range in which chaotic behavior occurs for the system is determined. The chaotic response of the system is regularized and controlled by applying an appropriate piezoelectric voltage which shifts the frequency response curve along the frequency axis.

A combined analysis of smooth and non-smooth bifurcations captures the interplay of different qualitative transitions in a canonical model of an impact pair, a forced capsule in which a ball moves freely between impacts on either end of the capsule. The analysis, generic for the impact pair context, is also relevant for applications. It is applied to a model of an inclined vibro-impact energy harvester device, where the energy is generated via impacts of the ball with a dielectric polymer on the capsule ends. While sequences of bifurcations have been studied extensively in single- degree-of-freedom impacting models, there are limited results for two-degree-of-freedom impacting systems such as the impact pair. Using an analytical characterization of impacting solutions and their stability based on the maps between impacts, we obtain sequences of period doubling and fold bifurcations together with grazing bifurcations, a particular focus here. Grazing occurs when a sequence of impacts on either end of the capsule are augmented by a zero-velocity impact, a transition that is fundamentally different from the smooth bifurcations that are instead characterized by eigenvalues of the local behavior. The combined analyses allow identification of bifurcations also on unstable or unphysical solutions branches, which we term ghost bifurcations. While these ghost bifurcations are not observed experimentally or via simple numerical integration of the model, nevertheless they can influence the birth or death of complex behaviors and additional grazing transitions, as confirmed by comparisons with the numerical results. The competition between the different bifurcations and their ghosts influences the parameter ranges for favorable energy output; thus, the analyses of bifurcation sequences yield important design information.

In this paper, a discrete-time predator–prey model with Crowley–Martin functional response is investigated based on the center manifold theorem and bifurcation theory. It is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation. An explicit approximate expression of the invariant curve, caused by Neimark–Sacker bifurcation, is given. The fractal dimension of a strange attractor and Feigenbaum’s constant of the model are calculated. Moreover, numerical simulations using AUTO and MATLAB are presented to support theoretical results, such as a cascade of period doubling with period-2, 4, 6, 8, 16, 32 orbits, period-10, 20, 19, 38 orbits, invariant curves, codimension-2 bifurcation and chaotic attractor. Chaos in the sense of Marotto is also proved by both analytical and numerical methods. Analyses are displayed to illustrate the effect of magnitude of interference among predators on dynamic behaviors of this model. Further the chaotic orbit is controlled to be a fixed point by using feedback control method.

In this paper, chaos of a new exponential logistic map modulated by Gaussian function is investigated. Firstly, the stability of the fixed point is analyzed, and the occurrence of period doubling bifurcation in the system is verified theoretically. Subsequently, the chaotic behavior of the system is analyzed using bifurcation diagrams, phase portraits, and Lyapunov exponents. The numerical results confirm the existence of chaos in the exponential logistic map within a specific parameter range. In addition, the proposed map has additional parameter degrees of freedom compared to the existing generalized logistic maps, which provides different chaotic characteristics and enhances design flexibility required for diverse applications. At last, we further study the two‐dimensional coupled exponential logistic map and find that the system enters chaos through two routes: period doubling bifurcation and Hopf bifurcation.

Stability and bifurcation conditions for a vibroimpact motion in an inclined energy harvester with T -periodic forcing are determined analytically and numerically. This investigation provides a better understanding of impact velocity and its influence on energy harvesting efficiency and can be used to optimally design the device. The numerical and analytical results of periodic motions are in excellent agreement. The stability conditions are developed in non-dimensional parameter space through two basic nonlinear maps based on switching manifolds that correspond to impacts with the top and bottom membranes of the energy harvesting device. The range for stable simple T -periodic behavior is reduced with increasing angle of incline β , since the influence of gravity increases the asymmetry of dynamics following impacts at the bottom and top. These asymmetric T -periodic solutions lose stability to period doubling solutions for β ≥ 0 , which appear through increased asymmetry. The period doubling, symmetric and asymmetric periodic motion are illustrated by bifurcation diagrams, phase portraits and velocity time series.

The paper studies bifurcations and complex dynamics in a class of nonautonomous oscillatory circuits with a flux-controlled memristor and harmonic forcing term. It is first shown that, as in the autonomous case, the state space of any memristor circuit of the class can be decomposed in invariant manifolds. It turns out that the memristor circuit dynamics is given by the collection of the dynamics of a family of circuits, with a nonlinear resistor in place of the memristor, which is parameterized by an additional constant input whose value depends on the initial conditions of the memristor circuit. This property makes it possible to employ the harmonic balance method in order to study the periodic solutions and their bifurcations due to changing the amplitude and the frequency of the harmonic input on a fixed manifold or due to changing the initial conditions for a fixed harmonic input. The main result is that in both of these cases the harmonic balance method is quite effective to accurately predict period doubling bifurcations of the periodic solutions. Analytical predictions are obtained in the cases of linear-plus-cubic and piecewise linear memristor flux–charge characteristics.

A discrete epidemic model with vaccination and limited medical resources is proposed to understand its underlying dynamics. The model induces a nonsmooth two dimensional map that exhibits a surprising array of dynamical behavior including the phenomena of the forward-backward bifurcation and period doubling route to chaos with feasible parameters in an invariant region. We demonstrate, among other things, that the model generates the above described phenomena as the transmission rate or the basic reproduction number of the disease gradually increases provided that the immunization rate is low, the vaccine failure rate is high and the medical resources are limited. Finally, the numerical simulations are provided to illustrate our main results.

The dynamics driven interaction between the bubbles in a cavitation cluster is known to be a complex phenomenon indicative of a highly active nonlinear as well as chaotic behavior in ultrasonic fields. By considering the cluster of encapsulated microbubble with a thin elastic shell in ultrasonic fields, in this paper, the dynamics of microbubbles has been studied via applying the methods of chaos physics. Bifurcation, Lyapunov exponent, and time series are plotted with respect to variables such as amplitude, initial bubble radius, frequency and viscosity. The findings of the study indicate that a bubble cluster undergoes a chaotic unstable region as the amplitude and frequency of ultrasonic pulse are increased mainly due to the period doubling phenomenon. The results of the present study are supported by findings of previous studies.

In this paper, we study the dynamics behaviour of a stratum of plant–herbivore which is modelled through the following F(x, y)=(f(x, y), g(x, y)) two-dimensional map with four parameters defined bywhere x≥0, y≥0, and the real parameters a, b, r, k are all positive. We will focus on the case a≠b. We study the stability of fixed points and do the analysis of the period-doubling and the Neimark–Sacker bifurcations in a standard way.