Narrow Results

We investigate the geodesic motion in D-dimensional Majumdar–Papapetrou multi-black hole spacetimes and find that the qualitative features of the D = 4 case are shared by the higher dimensional configurations. The motion of timelike and null particles is chaotic, the phase space being divided into basins of attraction which are separated by a fractal boundary, with a fractal dimension d_B. The mapping of the geodesic trajectories on a screen placed in the asymptotic region is also investigated. We find that the fractal properties of the phase space induces a fractal structure on the holographic screen, with a fractal dimension d_B - 1.

The dependence of the Casimir force on the optical properties and geometry of interacting materials makes possible to tailor the actuation dynamics of micro/nano actuators. In this research, we study the dynamical sensitivity of micro- and nanoelectromechanical systems on geometry by comparing the plate-plate and sphere-plate configurations, and taking into account the optical properties of the interacting materials. In fact, for conservative systems bifurcation analysis and phase portraits show that the geometry and the optical properties strongly influence the stability of an actuating device in a way that geometries that lead to weaker Casimir forces (sphere-plate geometry) favor more stable behavior. In addition, for non-conservative periodically driven systems, the Melnikov and Poincare portrait analysis shows that stronger Casimir forces lead to increased chaotic behavior, which more pronounced for the plate-plate geometry, that prohibits the long term prediction of the actuating dynamics of the system.

A semiclassical theory is used for the study of fidelity. For systems with weak chaos in the classical limit, we show that the fidelity has a non-Fermi-Golden-Rule decay, which can be explained by the closeness of the distribution of action difference in the semiclassical theory to the Levy distribution. For systems with strong chaos in the classical limit, we present a semiclassical expression for fidelity decay in the Lyapunov regime, which is more general than the previously predicted Lyapunov decay and λ₁ decay of fidelity. For systems with regular motion in the classical limit, we derive the fidelity decay for initial narrow Gaussian wavepackets, which displays a quite complex behaviour, from Gaussian to power law decay t^−α with 1 ≤ α ≤ 2.

In this paper, by employing both analytical and numerical methods, global dynamic responses including subharmonic bifurcations and chaos are investigated for a carbon nanotube supported by a Winkler and Pasternak foundation. The criteria of chaos arising from transverse intersections for stable and unstable manifolds of homoclinic orbits are proposed with the Melnikov method. The critical curves separating the chaotic and nonchaotic regions are plotted in the parameter plane. The parameter conditions for subharmonic bifurcations are also obtained by the subharmonic Melnikov method. It is proved rigorously that the route to chaos for this model is infinite subharmonic bifurcations. The stability of subharmonic bifurcations is also studied by the characteristic multipliers. Numerical simulations are given to confirm the analytical results.

A benchmark problem with air medium in a square cavity heated by a vertical side is studied in a Ra range near its critical value without using the usual model approximations, such as Boussinesq or low-Mach-number approximation. Two analysis methods of power spectrum and phase trajectory are adopted to distinguish the flow pattern in the cavity. It demonstrates how the flow evolves from transition to turbulent regimes with increasing Ra values.

Recent investigations of nonsmooth dynamical systems have resulted in the study of a class of novel bifurcations termed as sliding bifurcations. These bifurcations are a characteristic feature of so-called Filippov systems, that is, systems of ordinary differential equations (ODEs) with discontinuous right-hand sides. In this paper we show that sliding bifurcations also play an important role in organizing the dynamics of dry friction oscillators, which are a subclass of nonsmooth systems. After introducing the possible codimension-1 sliding bifurcations of limit cycles, we show that these bifurcations organize different types of "slip to stick-slip" transitions in dry friction oscillators. In particular, we show both numerically and analytically that a sliding bifurcation is an important mechanism causing the sudden jump to chaos previously unexplained in the literature on friction systems. To analyze such bifurcations we make use of a new analytical method based on the study of appropriate normal form maps describing sliding bifurcations. Also, we explain the circumstances under which the theory of so-called border-collision bifurcations can be used in order to explain the onset of complex behavior in stick-slip systems.

An analysis on the nonlinear forced vibration of thermally loaded FGM plate with two simply supported opposite and two free edges subjected to the in-plane and transversal excitations is presented. The material properties of the FGM plates are assumed to be temperature dependent and change continuously throughout the thickness of the plate, according to the volume fraction of the constituent materials based on the power law function. The temperature is assumed to be constant in the plane and varied only in the thickness direction of the plate. The plate is modeled by using the Von Karman hypothesis and the equations of motion are obtained by using an energy approach. It is our aim to choose a suitable mode function to satisfy the first two modes of transverse nonlinear oscillations and the boundary conditions for the FGM rectangular plates with two simply supported opposite and two free edges. The equations of motion can be reduced into a two-degree-of-freedom nonlinear system of transverse motion under combined thermal and external excitations by using the Galerkin's method. By the numerical method, the nonlinear dynamical equations are analyzed to find the nonlinear responses of the FGM plate with two simply supported opposite and two free edges. Under certain conditions the periodic motions and the chaotic motions of the FGM plate are found. The bifurcation diagram demonstrates that for a certain geometric and material properties the chaotic responses of the plate exist as the transverse excitation changes. Moreover, numerical simulations also illustrate that the deflections of the nonlinear dynamic of the FGM rectangular plate are larger than that of the periodic motions.

An analysis on nonlinear oscillations and chaotic dynamics is presented for a simply-supported symmetric cross-ply composite laminated rectangular thin plate with parametric and forcing excitations in the case of 1:3:3 internal resonance. Based on Reddy's third-order shear deformation plate theory and the von Karman-type equations, the nonlinear governing partial differential equations of motion for the composite laminated rectangular thin plate can be established via the Hamilton's principle. Such partial differential equations are further discretized by the Galerkin method to form a three-degree-of-freedom coupled nonlinear system including the cubic nonlinear terms. The method of multiple scales is then employed to derive a set of averaged equations. Through the stability analysis, the steady-state solutions of the averaged equations are provided. An illustrative case of 1:3:3 internal resonance and fundamental parametric resonance, 1/3 subharmonic resonance is considered. Numerical simulation is applied to investigate the intrinsically nonlinear behavior of the composite laminated rectangular thin plate. With certain external load excitations, the simulation results demonstrate that the nonlinear dynamical system of the composite laminated plate exhibits different kinds of periodic and chaotic motions.

Nonlinear dynamic behaviors of ceramic-metal graded truncated conical shell subjected to complex loads are investigated. The shell is modeled by first-order shear deformation theory. The nonlinear partial differential governing equation in terms of transverse displacements of the FGM truncated conical shell is derived from the Hamilton's principle. Galerkin's method is then utilized to discretize the partial governing equations to a two-degree-of-freedom nonlinear ordinary differential equation. The temperature-dependent materials properties of the constituents are graded in the radial direction in accordance with a power-law distribution. The aerodynamic pressure can be calculated by using the first-order piston theory. The temperature field is assumed to be a steady-state constant-temperature distribution. Bifurcation diagrams, the maximum Lyapunov exponents, wave forms and phase portraits are obtained by numerical simulation to demonstrate the complex nonlinear dynamics response of the FGM truncated conical shell. The influences of the semi-vertex angle, the material gradient index, in-plane and aerodynamic load on the nonlinear dynamics are studied.

A time-delayed quadratic nonlinear mechanical system can exhibit two coexisting stable bifurcating solutions (SBSs) after two-to-one resonant Hopf bifurcations occur in the corresponding autonomous time-delayed system. One SBS is of small-amplitude and has the Hopf bifurcation frequencies (HBFs), while the other is of large-amplitude and contains the shifted Hopf bifurcation frequencies (the shifted HBFs). When the forcing frequency is tuned to be the sum of two HBFs or the sum of two shifted HBFs, two families of additive resonances can be induced in the forced response. The forced response under the additive resonance related to the HBFs can demonstrate periodic, quasi-periodic and chaotic motion. On the contrary, the forced response under the additive resonance associated with the shifted HBFs may exhibit period-three periodic motion and quasi-periodic motion. Bifurcation diagrams, time trajectories, frequency spectra, phase portraits and Poincaré sections are presented to show periodic, quasi-periodic, and chaotic motion of the time-delayed nonlinear system under the two families of additive resonances.

We elucidate the orbital dynamics of a binary system of two magnetic dipoles, by utilizing the grid classification method. Our target is to unveil how the total energy (expressed through the Jacobi constant), as well as the ratio of the magnetic moments affect the character of the trajectories of the test particle. By integrating numerically large data sets of starting conditions of trajectories in different types of 2D maps, we manage to reveal the basins corresponding to bounded, close encounter and escape motion, along with the respective time scales of the phenomena.

The nonlinear dynamics of an underdamped sinusoidal potential system is experimentally and numerically studied. The system shows regular (nonchaotic) periodic motion when driven by a small amplitude ($F_1$) sinusoidal force (frequency $\omega =\frac{2\pi }{\tau }$). However, when the system is driven by a similarly small amplitude biharmonic force (frequencies $\omega$ and $\frac{\omega }{2}$ with amplitudes $F_1$ and $F_2$, respectively) chaotic motion appear as a function of amplitude ($F_2$) of the $\frac{\omega }{2}$-frequency component for a fixed $F_1\approx 0.22$. We investigate the effect of an additional constant force $\mathrm{\Delta }F$ on the dynamics of the system in the ($F_2-\mathrm{\Delta }F$) space. We find that $\mathrm{\Delta }F$ can cause chaotic motion to move to regular motion and regular motion can also become chaotic in certain ($F_2-\mathrm{\Delta }F$) domains.

This paper presents the analysis of flutter and nonlinear dynamics of an orthotropic composite laminated rectangular plate subjected to aerodynamic pressures and transverse excitation. The first-order linear piston theory is employed to model the air pressures. Based on Reddy’s third-order shear deformation plate theory and von Karman-type equation for the geometric nonlinearity, the nonlinear governing equations of motion are derived for the composite laminated rectangular plate by applying the Hamilton’s principle. The Galerkin method is utilized to discretize the partial differential governing equations to a set of nonlinear ordinary differential equations. The critical Mach number for occurrence of the flutter of the composite laminated plate is investigated by solving the eigenvalues problem. The relationship between the limit cycle oscillation and the critical Mach number is analyzed based on the nonlinear equations. The numerical simulation is conducted to study the influences of the transverse excitation on the nonlinear dynamics of the composited laminated plate. The numerical results, which include bifurcation diagram, phase plots and waveforms, demonstrate that there exist the bifurcation and chaotic motions of the composited laminated plate subjected to the aerodynamic pressures and the transverse excitation.

A mechanical model is proposed to describe the vibration of a taut inclined cable excited by a deck. The dynamic system is simplified into a one-degree-of-freedom nonlinear system with non-dimensional parameters by using the Galerkin and non-dimensional analysis methods. The periodicity ratio (PR) method is improved and applied to the parameter analysis for the cable system. The frequency ratio between the natural frequency of the cable and the vibration frequency of the deck and the amplitude of deck vibration are the main variable parameters, and nonlinear motion is diagnosed for a large range of parameters. Several forms of motion are distinguished, and the evolution between the forms is studied. Meanwhile, several types of parametric resonance are revealed in different ranges of parameters. The results indicate that the parametric resonance in the case of a 3:2 frequency ratio is significant and should be given more attention.

The investigation of hyperelastic responses of soft materials and structures is essential for understanding of the mechanical behaviors and for the design of soft systems. In this paper, by considering both the material and geometrical nonlinearities, a new neo-Hookean model for the hyperelastic beam is developed with focus on its nonlinear free vibration with large strain deformations. The neo-Hookean model is employed to capture the large strain deformation of the hyperelastic beam. The governing equations of the hyperelastic beam are derived by using Hamilton’s principle. To avoid expensive calculations for solving the nonlinear governing equations, a simplified Taylor-series expansion model is proposed. The effects of two key system parameters, i.e. the initial displacement amplitude and the slenderness ratio, on the nonlinear free vibrations of the hyperelastic beam are numerically analyzed. The bifurcation diagrams, displacement time traces, phase portraits and power spectral diagrams are presented for the nonlinear free vibrations of the hyperelastic beam. For small initial displacement amplitudes, it is found that the hyperelastic beam will undergo limit cycle oscillations, depending on the initial amplitude employed. For initial displacement amplitudes large enough, interestingly, the free vibration of the hyperelastic beam will become quasi-periodic or chaotic, which were rarely reported for the free vibration of linearly elastic beams. Also observed is the traveling wave feature of oscillating shapes of the hyperelastic beam, indicating that higher-order modes of the beam are excited even for free vibrations. All these new features in the nonlinear free vibrations of hyperelastic beams indicate that the material and geometric nonlinearities play a great role in the dynamic analysis of hyperelastic beams.

In this study, nonlinear vibration and stability of a pipe conveying fluid, with axially moving supports, immersed in an external fluid is studied. The equation of motion is derived with the aid of the Newtonian method via employing the Euler–Bernoulli beam theory and plug flow assumption. Considering the stretching effect, the nonlinear equation of motion is obtained, and it is discretized via the Galerkin method. Afterward, the stability of the system is investigated using two different approaches, and their results are compared. Numerical simulations show the system has complex dynamic behavior. To unveil the dynamics of the system, various analysis techniques such as bifurcation diagram of Poincaré map, phase portrait, power spectrum, and time trajectories are used. Limit cycle, period-n, quasi-periodic, and chaotic regime are observed. It is found that pipe can flutter around the trivial solution or post-buckling equilibrium points. Moreover, the influences of some system parameters on the chaotic behavior of the pipe are discussed.

In an attempt to improve the current low efficiency and high consumption situation of vibration mills, this paper analyses the chaotic motion characteristics of the system and the movement of vibration mill. The complex stiffness-dispersion coupling of the system is also studied, so as to investigate the effect of the system’s chaotic motion characteristics on the efficiency improvement and energy consumption reduction. Based on the ADAMS software, this paper establishes a simplified vibration mill mechanical model, analyzes the singularity and stability of the system, and determines the critical speed at which the vibration motor becomes chaotic according to the bifurcation diagram. Then the chaotic state of the grinding machine with sinusoidal variation in its motor speed is studied based on the Poincaré principle, singular attractor and maximum Lyapunov exponent. Lastly, a 200h vibration test on diamond powder with an average particle size of 10 $\mu$m was carried out. Test results under the two operating conditions of variable and constant speeds are compared and analyzed. Our results show that with variable speed the vibration mill achieved higher grinding efficiency but smaller particle grain size. The research elaborated in this paper provides a valuable reference for the engineering application of the chaotic characteristics of vibration mill.

A truncated conical shallow shell rotating around a single axle and subjected to a transverse periodic loading is studied in this research. The Melnikov method is adopted to study the homoclinic bifurcation and subharmonic bifurcation of the dynamic system. Numerical simulations are performed for investigating the system's nonlinear and chaotic behavior. Conditions under which chaotic motions occur are determined.

We study the phase space dynamics of multi–dimensional symplectic maps, using the method of the Generalized Alignment Index (GALI). In particular, we investigate the behavior of the GALI for a system of N = 3 coupled standard maps and show that it provides an efficient criterion for rapidly distinguishing between regular and chaotic motion.

In this paper, we present a study of the dynamical behavior in a Kelvin type gyrostat satellite. We firstly obtain the Hamiltonian equations of our model by using Cardan angles as generalized coordinates. Then, we make this Hamiltonian dimensionless and calculate motion equations for this dimensionless system. The study of the Poincare's sections of this system shows us that chaotic motion regimes are present for specific parameter values. The main goal of this work is the finding of stabilizing orbits by using a control technique, the fuzzy control of Poincare map method, so that it can be applied to stabilize special periodic orbits in this system. Finally, we expect that the technique can be useful for a better understanding of control theory and their applications in gyrostat problems.