Papers by Keyword: Bifurcation

Abstract: This paper reports a novel three-dimensional autonomous chaotic system. By choosing an appropriate bifurcation parameter, we prove that a Hopf bifurcation occurs in this system when the bifurcation parameter exceeds a critical value, and some basic dynamical properties, such as Lyapunov exponents, fractal dimension, bifurcation diagram, continuous spectrum and chaotic dynamical behaviors of the new chaotic system are studied. Furthermore, the forming mechanism of its compound structure obtained by merging together two simple attractors after performing one mirror operation has been investigated by detailed numerical as well as theoretical analysis.

Abstract: Longitudinal vibration of a nonlinear viscoelastic rod system with one end fixed and another end subjected to an axial periodical excitation was studied under the consideration of transverse inertia. By using Galerkin method, a combined Parametric and Forcing Excited cubic nonlinear dynamic system is derived for hard stiffness nonlinear material. Furthermore, arc-length technique is used for an accurate integral procedure, and numeric results are given detailedly. The process of the system evolved from stable periodic motion to chaos is illustrated in a period-doubling bifurcation graph in a parameter space, and the Lyapunov exponent spectrum is also given that is perfectly consistent with bifurcation process. The strange attractor obtained from Poincaré Map is present, which has different fractal dimension from Duffing’s one, so it may be a new chaotic attractor.

Abstract: Using the analytical and numerical approaches, the nonlinear dynamic behaviors in the vicinity of a compound critical point are studied for a simply supported functionally graded materials (FGMs) rectangular plate. Normal form theory, bifurcation and stability theory are used to find closed form solutions for equilibria and periodic motions. Stability conditions of these solutions are obtained explicitly and critical boundaries are also derived. Finally, numerical results are presented to confirm the analytical predictions

Abstract: The homoclinic tangency for a rotor-active magnetic bearings (AMB) system with the time-varying stiffness are considered in this paper. The zeros of Melnikov equation are paid more attentions and a 3-order zero was gained and some numerical results under the parameter perturbations were shown.

Abstract: This paper is concerned with the dynamical behavior of a chaotic system which is a model for seismic response of structures. The local bifurcation of the non-hyperbolic equilibrium point of the chaotic system is investigated by using center manifold method. The transcritical bifurcation is analyzed in detail. Based on numerical simulations, spectrums of maximal Lyapunov exponent and the bifurcation diagrams are presented for the dynamic analysis. The method proposed can be used as a reference of nonlinear seismic response analysis.

Abstract: The spur gear pair’s nonlinear equation of motion including piece-wise backlash and internal error excitation is derived in this research. The worn tooth effect in time-varying mesh stiffness is introduced to do in-depth investigation of the dynamic traits for gear transmission system with wear fault. The internal excitation frequency is selected as a criterion to calculate the bifurcation diagram and the corresponding Lyapunov exponents. Some auxiliary analyzing meanings such as Poincaré maps, phase trajectory, power spectrum and time history curve are utilized to illustrate the system’s nonlinear behaviors with special parameter settings. Different routes to chaos and abundant nonlinear phenomena have been observed in this nonlinear gear transmission system.

Abstract: In this paper, the pure rotational dynamic model of one stage gear pair system is developed and the nonlinear factors, such as strong nonlinear suspension, time-varying mesh stiffness, piece-wise backlash and internal error excitation, are included in our model. The relative mesh displacement is employed to convert the semi-define system with rigid displacement model into a define one. Such qualitative and quantitative methods as the bifurcation diagram, Lyapunov exponents, Poincaré section, phase trajectory, power spectrum and time history curve are utilized in present research to illustrate the nonlinear dynamic behavior of the system. The sub-harmonic response, chaotic response and the route to chaos of the system are revealed based on one applicable numerical integration method.

Abstract: Use the same architectural language, different expression technique to meet the function of the construction and its own aesthetic. Use braiding, weaving and bifurcation to express the form, patterns and structure. Because the function of the architecture is a fashion museum, so from internal to external fashion tangling lines.

Abstract: In this paper, the peakons and bifurcations in a generalized Camassa-Holm equation are studied by using the bifurcation method and qualitative theory of dynamical systems. First, the averaged equation is obtained by introducing linear transform and traveling wave transform to the generalized Camassa-Holm equation. Then, we applied the bifurcation theory of planar dynamical system and maple software to investigate the averaged equation. The phase portrait of the system under a parameter condition is obtained. Finally, we get the peakons from the limit of general single solitary wave solution.

Abstract: Based on the flow model and its mathematical description in curvilinear coordinates, the steady and fully developed laminar flow in the curved pipe of square cross-section was simulated by the Galerkin method. The results show that the eye points of the secondary flow vortex at low Reynolds numbers almost locate at the position of 0.457 on the vertical axis, the axial velocity maximum shifts inward or outward when the pipe curvature parameter and flow conditions change. Furthermore the bifurcation flow is studied to draw the bifurcation state diagram, and the Reynolds number corresponding to the bifurcation point of the four-vortex solution is accurately equal to 300.1.