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This paper presents an analysis of the chaotic motion and its control for the nonlinear nonplanar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitations at the free end. A new method of controlling chaotic motion for the nonlinear nonplanar oscillations of the cantilever beam, refereed as to the force control approach, is proposed for the first time. The governing nonlinear equations of nonplanar motion under combined parametric and external excitations are obtained. The Galerkin procedure is applied to the governing equation to obtain a two-degree-of-freedom nonlinear system under combined parametric and forcing excitations for the in-plane and out-of-plane modes. The work is focused on the case of 2:1 internal resonance, principal parametric resonance-1/2 subharmonic resonance for the in-plane mode and fundamental parametric resonance-primary resonance for the out-of-plane mode. The method of multiple scales is used to transform the parametrically and externally excited system to the averaged equations which have a constant perturbation force. Based on the averaged equations obtained here, numerical simulation is utilized to discover the periodic and chaotic motions for the nonlinear nonplanar oscillations of the cantilever beam. The numerical results indicate that the transverse excitation in the z direction at the free end can control the chaotic motion to a period n motion or a static state for the nonlinear nonplanar oscillations of the cantilever beam. The methodology of controlling chaotic motion by using the transverse excitation is proposed. The transverse excitation in the z direction at the free end may be thought about to be an open-loop control. For the problem investigated in this paper, this approach is an effective methodology of controlling chaotic motion to a period n motion or a static state for the nonlinear nonplanar oscillations of the cantilever beam.

This paper presents an approach to characterize the conditions that can possibly lead to chaotic motion for a simply supported large deflection circular plate of thermo-mechanical coupling by utilizing the criterion of the maximum Lyapunov exponent. The governing partial differential equation of the simply supported large deflection circular plate of thermo-mechanical coupling is first derived and simplified to a set of three ordinary differential equations by the Galerkin method. Several different features including time history, Power spectra, phase plot, Poincare map and bifurcation diagram are then numerically computed. These features are used to characterize the dynamic behavior of the plate subjected various geometric and excitation conditions. Numerical examples are presented to verify the validity of the conditions that lead to chaotic motion and the effectiveness of the proposed modeling approach. The modeling results of numerical simulation indicate that the chaotic motion may occurs in the lateral loads f =2.5 , η1=1.1, β=0.5, δ =0.01 and e∞=0.0007. As the thermo-elastic damping is great than a critical value, the dynamic motion of the thermal-couple plate is periodic. As the thickness parameter β of the concave circular plate is great than a critical value, the motion of the plate is periodic. The modeling result thus obtained by using the method proposed in this paper can be employed to predict the instability induced by the dynamics of the thermo-mechanical coupling circular plate in large deflection.

This paper presents a new approach to characterize the conditions that can possibly lead to chaotic motion for a simply supported large deflection rectangular plate of thermo-mechanical coupling by utilizing the criteria of the fractal dimension and the maximum Lyapunov exponent. The governing partial differential equation of the simply supported rectangular plate of thermo-mechanical coupling is first derived and simplified to a set of three ordinary differential equations by the Galerkin method. Several different features including power spectra, phase plot, Poincare map and bifurcation diagram are then numerically computed. These features are used to characterize the dynamic behavior of the plate subjected to various excitation conditions. Numerical examples are presented to verify the validity of the conditions that lead to chaotic motion and the effectiveness of the proposed modeling approach. The numerical results indicate that large deflection motion of the thermo-mechanical coupling rectangular plate possesses many bifurcation points, chaotic motions and period-double phenomena under various lateral loads, bi-axial loads, thermo-mechanical coupling factors and aspect ratios. The modeling results of numerical simulation indicate that the chaotic motion may occur in the range of lateral loads Q =1.25 to 3.35 and near the bi-axial load η2=1.5. The dynamic motion of the thermal-couple plate is periodic if the aspect ratio is within a specific range or the thermo-mechanical coupling factor is within a specific range. The modeling result thus obtained by using the method proposed in this paper can be employed to predict the instability induced by the dynamics of the thermo-mechanical coupling plate in large deflection.

In this paper, we formulate the intricate process of Human Immunodeficiency Virus (HIV) infection in a mathematical model. In the formulation of the model, the density of CD 4 + T-cells is categorized into healthy and infected classes while the density of free HIV viruses in the human host’s blood is considered a separate class. It is reported that when HIV enters a human’s body, it infects a large amount of CD 4 + T-cells and unbalance the supply of new cells from the thymus. Therefore, we incorporate variable source term relying on the viral load for the supply of new cells instead of using a fixed value for the supply of these cells. We implement a continuous Galerkin Petrov time discretization scheme, particularly cGP(2)-scheme to visualize the dynamical behavior of the mentioned model and a detailed description of the effects of different physical parameters of interest are depicts graphically and discuss that how the level of healthy, infected CD 4 + T-cells and free HIV particles varies concerning the emerging parameters in the model. In the proposed cGP(2)-method, two unknowns terms can be found by solving a 2 × 2 block system. This method is accurate of order 3 in the whole time interval and shows even superconvergence of order 4 in the discrete-time points. Furthermore, determine the solution of the model utilizing the fourth-order Runge-Kutta (RK4)-method and find out the absolute errors between the solutions obtained from both approaches. Finally, the validity and reliability of the proposed scheme are verified by comparing the numerical and graphical results with the RK-4 method. We examine the accuracy of the cGP(2)-sheme and observe that it produces more accurate results at relatively larger step sizes as in comparison to RK-4 scheme. Correspondence between the findings of cGP(2)-method and RK4-method reveal that the new technique is a promising tool for obtaining approximate solutions to the nonlinear systems of real-world problems.

Recent advances in the field of uncertainty quantification are based on achieving suitable functional representations of the solutions to random systems. This aims at improving the performance of Monte Carlo simulation, at least for low-dimensional problems and moderately large independent variable. One of these functional representations are the so-called generalized polynomial chaos (gPC) expansions, based upon orthogonal polynomial decompositions. When the input random parameters are independent (a germ), a Galerkin projection technique applied to the truncated gPC expansion is usually employed. This approach exhibits fast mean-square convergence for smooth dynamics, whenever applicable. However, the main difficulty arises when solving the Galerkin system for the gPC coefficients, which may rely on different solvers (algorithms and codes) than those for the original system. A recent contribution noticed that, for random Hamiltonian systems, the Galerkin system is Hamiltonian too. Thus, the well-known symplectic integrators can be applied. The present paper investigates random Hamiltonian systems in general, when the input random parameters may be non-independent. In such a case, polynomial expansions based on the canonical basis and an imitation of the Galerkin projection technique are proposed. The Hamiltonian structure of the original system is unfortunately not conserved, but volume preservation is. Hence volume-preserving integrators are of use. Numerical experiments suggest that the proposed polynomial expansions may be useful for fast and accurate uncertainty quantification.

With analytical and numerical methods, dynamic response of a composite laminated buckled beam with a lumped mass and an axial excitation is investigated in this paper. The dynamic equation of the first mode for this model, which is a perturbation of a Hamilton system with both quadratic and cubic nonlinearities as well as external and parametric excitations, is obtained with the Galerkin method. The distribution of the equilibriums, homoclinic orbits and periodic orbits for the unperturbed system under different system parameters are obtained analytically. Chaotic vibrations arising from homoclinic intersections are studied with Melnikov method. Chaotic feature on the location of the lumped mass is investigated in detail. It is presented that there exist controllable frequencies decreasing monotonously with the location of the lumped mass for this system. Subharmonic bifurcation is also studied with the subharmonic Melnikov method. It is numerically proved that the system may undergo chaotic motions through finite subharmonic bifurcations. Numerical simulations including the phase portraits, time histories, Poincar e ´ sections, Lyapunov spectrums and bifurcation diagram are given, which confirm the analytical results.

In this paper, a nonlinear Euler-Bernoulli beam under a concentrated harmonic excitation with intermediate nonlinear support is investigated. Continuous expression for the kinetic energy, potential energy and dissipation function are constructed. An energy method based on the Lagrange equation combined with the Galerkin truncation is used for discretizing the governing equation. The Multi-dimensional incremental harmonic balance method (MIHBM) is derived, and the comparisons between the numerical results and the approximate analytical solutions based on the MIHBM verify the excellent accuracy of the MIHBM. The steady state dynamic of the beam is investigated by MIHBM. In order to investigate the energy transmission and understand the vibration response of the Euler-Bernoulli beam, the effects of the key parameters on the dynamic behaviors are studied and discussed, individually. The results show that the amplitude-frequency curves exhibits softening nonlinear behavior in the super-harmonic resonance region, and near resonant region the hardening nonlinear behavior is observed depending on the different parameters. Nonlinear dynamic analysis, such as bifurcation, 3-D frequency spectrum, waveform, frequency spectrum, phase diagram and Poincare map, are also presented in order to study the influences of the key parameters on the vibration behaviors for the beam in a more accurate manner. In addition, the path to chaotic motion is observed to be through a sequence of the periodic motion and quasi-periodic motion.

The Fractional-order model of fin problem in the presence of a temperature-dependent rate of internal heat generation in fin material has been studied in the proposed paper. Fractional-order Legendre functions (F-OLFs) with the Galerkin approach has been used to compute the result of the proposed fin problem. The exact solution is also calculated in a particular case of our fractional-order fin problem. The proposed fin problem has been presented in the non-dimension form. Temperature distribution in the straight fin is calculated for different values of used variables or parameters along space coordinate. The impact of different parameters such as fractional order, fin parameter, internal heat generation, generation number, etc. on the temperature profile in the fin is described in detail. The cooling process is faster in fractional order fin problem than the integer-order.

This paper investigated the nonlinear dynamics of a cracked vibration platform subjected to a concentrated transverse excitation force considering the structural and supported nonlinearities. To precisely predict vibration properties of the continuum structure, the dynamic model is established based on mechanical relations and the governing differential equations of motion are obtained with the nonlinear regime via an energy method according to the Lagrange's equation, which are discretized by utilizing the Galerkin truncation method. A comparative analysis via the finite element method (FEM) for verification accuracy and convergence is performed. To investigate the influences of the key parameters on the nonlinear behaviors of cracked platform, a wide range of excitation, nonlinear support and crack parameters are analyzed and discussed by bifurcation, time domain for nonlinear mechanics analysis. The numerical results show that a considerable effect of excitation amplitude on the global dynamics for the vibration platform is discovered, namely the higher excitation is able to suppress the chaotic behaviors but increase the vibration amplitude of the system. The increasing nonlinear support stiffness may enhance the chaotic characteristics of the vibration platform, the support position can effectively control the chaotic motion and nonlinear vibration as the support is close to the middle of vibration platform. The crack parameters play an effective role in dynamic behaviors.

The global bifurcations and chaotic dynamics of a string-beam coupled system subjected to parametric and external excitations are investigated in detail in this paper. The governing equations are firstly obtained to describe the nonlinear transverse vibrations of the string-beam coupled system. The Galerkin procedure is introduced to simplify the governing equations of motion to ordinary differential equations with two-degrees-of-freedom. Using the method of multiple scales, parametrically and externally excited system is transformed to the averaged equation. The case of 1:2 internal resonance between the modes of the beam and string, principal parametric resonance for the beam and primary resonance for the string is considered. Based on the averaged equation, the theory of normal form is utilized to find the explicit formulas of normal form associated with one double zero and a pair of pure imaginary eigenvalues. The global perturbation method is employed to analyze the global bifurcations and chaotic dynamics of the string-beam coupled system. The analysis of the global bifurcations indicates that there exist the homoclinic bifurcations and the Silnikov type single-pulse homoclinic orbit in the averaged equation of the string-beam coupled system. These results obtained here mean that the chaotic motions can occur in the string-beam coupled system. Numerical simulations also verify the analytical predications.

The attitude stability of flexible spacecraft is discussed in this paper. The Euler’s equations of spacecraft and the equations of forced vibration of two flexible solar arrays are derived. The governing equation is discretized via the Galerkin method, and the stability of relative equilibrium of spacecraft in the orbit frame is determined based on the Kelvin–Tait–Chetayev theorem. The sufficient condition of stability in the analytical form for the n -term truncated modal is established based on the governing equation.

This paper is devoted to the investigation of bifurcation and chaos of a moving viscoelastic string undergoing three-dimensional vibration. An initially stressed viscoelastic string subjected to constant axial motion and harmonical tension is considered. The string material is assumed to be constituted by spring-dashpot model which is considered as the Voigt element in series with a spring (three-parameter model). Using constitutive law of differential type method we can obtain the partial differential equations of motion. Then the partial differential equations can be approximated by means of Galerkin’s method. The axial–lateral coupled effect of the viscoelastic string is digged out by comparing the numerical results of the three-dimensional vibration analysis and the only transverse vibration analysis. The responses of the viscoelastic string and the elastic string are compared. Using the FFT spectra, Poincare maps, and the bifurcation diagrams, the responses of the viscoelastic string are classified. The effects of the material parameters, the wave propagation speed ratio, the transport speed and the frequency of axially tension on the response amplitudes are investigated.

In this paper, the Shilnikov type multi-pulse orbits and chaotic dynamics of parametrically excited viscoelastic moving belt are studied in detail. Using Kelvin-type viscoelastic constitutive law, the equations of motion for viscoelastic moving belt with the external damping and parametric excitation are given. The four-dimensional averaged equation under the case of primary parametric resonance is obtained by directly using the method of multiple scales and Galerkin’s approach to the partial differential governing equation of viscoelastic moving belt. From the averaged equations obtained here, the theory of normal form is used to give the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues. Based on normal form, the energy-phrase method is employed to analyze the global bifurcations and chaotic dynamics in parametrically excited viscoelastic moving belt. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type multi-pulse homoclinic orbits in the averaged equation. The results obtained above mean the existence of the chaos for the Smale horseshoe sense in parametrically excited viscoelastic moving belt. The chaotic motions of viscoelastic moving belts are also found by using numerical simulation. A new phenomenon on the multi-pulse jumping orbits is observed from three-dimensional phase space.

This paper investigates bifurcation and chaos in transverse motion of axially accelerating viscoelastic beams. The Kelvin model is used to describe the viscoelastic property of the beam material, and the Lagrangian strain is used to account for geometric nonlinearity due to small but finite stretching of the beam. The transverse motion is governed by a nonlinear partial-differential equation. The Galerkin method is applied to truncate the partial-differential equation into a set of ordinary differential equations. When the Galerkin truncation is based on the eigenfunctions of a linear non-translating beam subjected to the same boundary constraints, a computation technique is proposed by regrouping nonlinear terms. The scheme can be easily implemented in practical computations. When the transport speed is assumed to be a constant mean speed with small harmonic variations, the Poincare map is numerically calculated based on 4-term Galerkin truncation to identify dynamical behaviors. The bifurcation diagrams are present for varying one of the following parameter: the axial speed fluctuation amplitude, the mean axial speed and the beam viscosity coefficient, while other parameters are unchanged.

This paper presents a novel analytical approach utilizing fractal dimension criteria and the maximum Lyapunov exponent to characterize the conditions which can potentially lead to the chaotic motion of a simply supported thermo-mechanically coupled orthotropic rectangular plate undergoing large deflections. The study commences by deriving the governing partial differential equations of the rectangular plate, and then applies the Galerkin method to simplify these equations to a set of three ordinary differential equations. The associated power spectra, phase plots, Poincare map, maximum Lyapunov exponents, and fractal and bifurcation diagrams are computed numerically. These features are used to characterize the dynamic behavior of the orthotropic rectangular plate under various excitation conditions. The maximum Lyapunov exponents and the correlation dimensions method indicate that chaotic motion of the orthotropic plate occurs at η1 = 1.0, p ¯ = 2.0 , and δ ¯ = 0.05 for an external force of f ¯ = 5.0 . The application of an external in-plane force of magnitude p ¯ = 1.08 causes the orthotropic plate to perform bifurcation motion. Furthermore, when p ¯ > 3.0 , aperiodic motion of the plate is observed. Hence, the dynamic motion of a thermo-mechanically coupled orthotropic rectangular plate undergoing large deflections can be controlled and manipulated to achieve periodic motion through an appropriate specification of the system parameters and loads.

In this paper, nonlinear dynamical behaviors of axially moving viscoelastic strings are investigated. The one-term and the two-term Galerkin truncations using translating string eigenfunctions are respectively employed to reduce the partial-differential equation that governs the transverse motions of the string to a set of ordinary differential equations. The bifurcation diagrams are presented in the case that the amplitude of the periodic perturbation, or the dynamic viscosity is respectively varied while other parameters are fixed. The dynamical behaviors are numerically identified based on the Poincare maps. Numerical results show that regular and chaotic motions occur in the transverse vibration of the axially moving viscoelastic strings.

This paper investigates bifurcation and chaos of an axially accelerating viscoelastic beam. The Kelvin–Voigt model is adopted to constitute the material of the beam. Lagrangian strain is used to account for the beam's geometric nonlinearity. The nonlinear partial–differential equation governing transverse motion of the beam is derived from the Newton second law. The Galerkin method is applied to truncate the governing equation into a set of ordinary differential equations. By use of the Poincare map, the dynamical behavior is identified based on the numerical solutions of the ordinary differential equations. The bifurcation diagrams are presented in the case that the mean axial speed, the amplitude of speed fluctuation and the dynamic viscoelasticity is respectively varied while other parameters are fixed. The Lyapunov exponent is calculated to identify chaos. From numerical simulations, it is indicated that the periodic, quasi-periodic and chaotic motions occur in the transverse vibrations of the axially accelerating viscoelastic beam.

This paper investigates chaotic behaviors of an axially traveling viscoelastic string with geometric nonlinearity. The stress and the strain of the viscoelastic string obey the Boltzmann superposition principle. The Galerkin method is applied to truncate a nonlinear partial-differential–integral equation governing transverse motion into a set of ordinary differential–integral equations. For the string modeled as a standard linear solid, new auxiliary variables are introduced to transform those equations into ordinary differential equations. By use of the Poincare maps, the chaotic behaviors are presented based on the numerical solutions of the ordinary differential equations. The bifurcation diagrams are presented for varying one of the following parameter: the axial traveling speed, the amplitude of tension fluctuation, the viscoelastic exponent and coefficient of the string, while other parameters are fixed.

Non-linear vibrations of beam-like model of two-dimensional axially moving web with time-dependent tension have been investigated in this paper. The beam model material as the Kelvin–Voigt element is considered. The Galerkin method and the fourth-order Runge–Kutta method were used to solve the governing non-linear partial–differential equation. The effects of the transport speed, the tension perturbation amplitude and the internal damping on the dynamic behaviour of the system were numerically investigated. The Poincare maps have been constructed to classify the vibrations. For small values of the transport speed and the amplitude of periodic perturbation the system is asymptotically stable with its response tending to zero. With the increase of parameters one can observe local pitchwork type bifurcation and the coexistence of attractors. For small values of internal damping chaotic motion occurs. Regular and chaotic motions occur when internal damping increases.

This paper presents the conditions that can possibly lead to chaotic motion and bifurcation behavior for a simply-supported large deflection thermo-elastic circular plate with variable thickness by utilizing the criteria of fractal dimensions, maximum Lyapunov exponents and bifurcation diagrams. The governing partial differential equation of the simply supported thermo-elastic circular plate with variable thickness is first derived by means of Galerkin method. Several different features including Fourier spectra, phase plot, Poincar’e map and bifurcation diagrams are numerically computed. These features are used to characterize the dynamic behavior of the plate subjected to various excitations of lateral loads and thermal loads. Numerical examples are presented to verify the conditions that lead to chaotic motion and the effectiveness of the proposed modeling approach. Numerical modeling results indicate that large deflection motion of a thermo-elastic circular plate with variable thickness possesses chaotic motions and bifurcation motion under different lateral loads and thermal loads. The simulation results also indicate that the periodic motion of a circular plate can be obtained for the convex or the concave circular plate. The dynamic motion of the circular plate is periodic for the cases including (1) the lateral loading frequency is within a specific range, (2) thermal and lateral loadings are operated in a specific range and (3) the thickness parameter is less than a specific critical value for the convex circular plate or greater than a specific critical value for the concave circular plate. The modeling results show that the proposed method can be employed to predict the non-linear dynamics of any large deflection circular plate with variable thickness.

The present paper is aimed to display the performance of the Fourier-Galerkin technique developed earlier by the authors for numerical treatment of localized solutions in L 2(-∞, ∞) for higher-order equations. The solitary-wave solution of the Kortweg-de Vries fifth order equation is obtained numerically and compared with approximate results of other authors.