Stability and local bifurcation analyses of two-wheeled trailers considering the nonlinear coupling between lateral and vertical motions

The nonlinear dynamics of two-wheeled trailers is investigated using a spatial 4-DoF mechanical model. The non-smooth characteristics of the tire forces caused by the detachment of the tires from the ground and other geometrical nonlinearities are taken into account. Beyond the linear stability analysis, the nonlinear vibrations are analyzed with special attention to the nonlinear coupling between the vertical and lateral motions of the trailer. The center manifold reduction is performed leading to a normal form up to third degree terms. The nature of the emerging periodic solutions, and, thus, the sense of the Hopf bifurcations are verified semi-analytically and numerically. Simplified models of the trailer are also used in order to point out the practical relevance of the study. It is shown that the linearly independent pitch motion affects the sense of the Hopf bifurcations at the linear stability boundary. Namely, the constructed spatial trailer model provides subcritical bifurcations for higher center of gravity positions, while the commonly used simplified mechanical models explore the less dangerous supercritical bifurcations only. Domains of loss of contact of tires are also detected and shown in the stability charts highlighting the presence of unsafe zones. Experiments are carried out on a small-scale trailer to validate the theoretical results. A good agreement can be observed between the measured and numerically determined critical speeds and vibration amplitudes.

Bifurcation Analysis and H∞ Control of a Stochastic Competition Model with Time Delay

In this paper, we study a stochastic competition model with time delay and harvesting.We first simplify it through the stochastic center manifold reduction principle and stochastic averaging method as a one-dimensional Markov diffusion process. Singular boundary theory and an invariant measure are applied to analyze stochastic stability and bifurcation. The T-S fuzzy model of the system is constructed, and the H∞ fuzzy controller is designed to eliminate the bifurcation phenomenon through a linear matrix inequality approach. Numerical simulation is used to demonstrate our results.

Feedback control of a nonlinear aeroelastic system with non-semi-simple eigenvalues at the critical point of Hopf bifurcation

The feedback control of Hopf bifurcation of nonlinear aeroelastic systems with asymmetric aerodynamic lift force and nonlinear elastic forces of the airfoil is discussed. For the Hopf bifurcation analysis, the eigenvalue problems of the state matrix and its adjoint matrix are defined. The Puiseux expansion is used to discuss the variations of the non-semi-simple eigenvalues, as the control parameter passes through the critical value to avoid the difficulty for computing the derivatives of the non-semi-simple eigenvalues with respect to the control parameter. The method of multiple scales and center-manifold reduction are used to deal with the feedback control design of a nonlinear system with non-semi-simple eigenvalues at the critical point of the Hopf bifurcation. The first order approximate solutions are developed, which include gain vector and input. The presented methods are based on the Jordan form which is the simplest one. Finally, an example of an airfoil model is given to show the feasibility and for verification of the present method.

Time delay induced Hopf bifurcation in a diffusive predator–prey model with prey toxicity

In this paper, we consider a diffusive predator–prey model with a time delay and prey toxicity. The effect of time delay on the stability of the positive equilibrium is studied by analyzing the eigenvalue spectrum. Delay-induced Hopf bifurcation is also investigated. By utilizing the normal form method and center manifold reduction for partial functional differential equations, the formulas for determining the property of Hopf bifurcation are given.

Hopf bifurcation in a delayed reaction–diffusion–advection equation with ideal free dispersal

In this paper, we investigate a delay reaction–diffusion–advection model with ideal free dispersal. The stability of positive steady-state solutions and the existence of the associated Hopf bifurcation are obtained by analyzing the principal eigenvalue of an elliptic operator. By the normal form theory and the center manifold reduction, the stability and bifurcation direction of Hopf bifurcating periodic solutions are obtained. Moreover, numerical simulations and a brief discussion are presented to illustrate our theoretical results.

Reduction principle at work

The purpose of this informal paper is three-fold: First, filling a gap in the literature, we provide a (necessary and sufficient) principle of linearized stability for nonautonomous difference equations in Banach spaces based on the dichotomy spectrum. Second, complementing the above, we survey and exemplify an ambient nonautonomous and infinite-dimensional center manifold reduction, that is Pliss’s reduction principle suitable for critical stability situations. Third, these results are applied to integrodifference equations of Hammerstein- and Urysohn-type both in C- and Lp -spaces. Specific features of the nonautonomous case are underlined. Yet, for the simpler situation of periodic time-dependence even explicit computations are feasible.

Hopf bifurcation in a reaction-diffusive-advection two-species competition model with one delay

In this paper, we investigate a reaction-diffusive-advection two-species competition model with one delay and Dirichlet boundary conditions. The existence and multiplicity of spatially non-homogeneous steady-state solutions are obtained. The stability of spatially nonhomogeneous steady-state solutions and the existence of Hopf bifurcation with the changes of the time delay are obtained by analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized system. By the normal form theory and the center manifold reduction, the stability and bifurcation direction of Hopf bifurcating periodic orbits are derived. Finally, numerical simulations are given to illustrate the theoretical results.

A diffusive predator-prey model with generalist predator and time delay

<abstract><p>Time delay in the resource limitation of the prey is incorporated into a diffusive predator-prey model with generalist predator. By analyzing the eigenvalue spectrum, time delay inducing instability and Hopf bifurcation are investigated. Some conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution are obtained by utilizing the normal form method and center manifold reduction for partial functional differential equation. The results suggest that time delay can destabilize the stability of coexisting equilibrium and induce bifurcating periodic solution when it increases through a certain threshold.</p></abstract>

Stability and dynamic transition of vegetation model for flat arid terrains

<p style='text-indent:20px;'>In this paper, we aim to investigate the dynamic transition of the Klausmeier-Gray-Scott (KGS) model in a rectangular domain or a square domain. Our research tool is the dynamic transition theory for the dissipative system. Firstly, we verify the principle of exchange of stability (PES) by analyzing the spectrum of the linear part of the model. Secondly, by utilizing the method of center manifold reduction, we show that the model undergoes a continuous transition or a jump transition. For the model in a rectangular domain, we discuss the transitions of the model from a real simple eigenvalue and a pair of simple complex eigenvalues. our results imply that the model bifurcates to exactly two new steady state solutions or a periodic solution, whose stability is determined by a non-dimensional coefficient. For the model in a square domain, we only focus on the transition from a real eigenvalue with algebraic multiplicity 2. The result shows that the model may bifurcate to an <inline-formula><tex-math id="M1">\begin{document}$ S^{1} $\end{document}</tex-math></inline-formula> attractor with 8 non-degenerate singular points. In addition, a saddle-node bifurcation is also possible. At the end of the article, some numerical results are performed to illustrate our conclusions.</p>