Analysis of Geometric Invariants for Three Types of Bifurcations in 2D Differential Systems

Little is known about bifurcations in two-dimensional (2D) differential systems from the viewpoint of Kosambi–Cartan–Chern (KCC) theory. Based on the KCC geometric invariants, three types of static bifurcations in 2D differential systems, i.e. saddle-node bifurcation, transcritical bifurcation, and pitchfork bifurcation, are discussed in this paper. The dynamics far from fixed points of the systems generating bifurcations are characterized by the deviation curvature and nonlinear connection. In the nonequilibrium region, the nonlinear stability of systems is not simple but involves alternation between stability and instability, even though systems are invariably Jacobi-unstable. The results also indicate that the dynamics in the nonequilibrium region are node-like for three typical static bifurcations.

A geometric interpretation of maximal Lyapunov exponent based on deviation curvature

In this study, we discuss a relationship between the behavior of nonlinear dynamical systems and geometry of a system of second-order differential equations based on the Jacobi stability analysis. We consider how a maximal Lyapunov exponent is related to the geometric quantities. As a result of a theoretical investigation, the maximal Lyapunov exponent can be represented by a nonlinear connection and a deviation curvature. Thus, this means that the Jacobi stability given by the sign of the deviation curvature affects the change of the maximal Lyapunov exponent. Additionally, for an equation of nonlinear pendulum, we numerically confirm the theoretical results. We observe that a change of the maximal Lyapunov exponent is related to a change of an average deviation curvature. These results indicate that the deviation curvature and Jacobi stability are essential for considering the change of maximal Lyapunov exponent.

Nonlinear Connection Stiffness Identification of Heritage Timber Buildings Using a Temperature-Driven Multi-Model Approach

‘Que-Ti’ is an important component in typical Tibetan heritage timber buildings and it performs similar to corbel brackets connecting beam and column in modern structures. It transfers shear, compression and bending moment by slippage and deformation of components as well as limited joint rotation. A rigorous analytical model of ‘Que-Ti’ is needed for predicting the behavior of a timber structure under extreme loadings. Few researches have been done on this topic, particularly with the parameters describing the performances of this connection subjected to external loads. In this paper, a new temperature-driven multimodel approach is proposed to identify the stiffness parameters of a ‘Que-Ti’ connection in its operating environment. Models with nonlinear compression and rotational springs have been developed to take into account the change of mechanical behavior of the ‘Que-Ti’ affected by the temperature variation in typical heritage Tibetan buildings. The column–beam connection is modeled as two nonlinear rotational springs and one nonlinear compressive spring. Ambient temperature variation is treated as a force function in the input (temperature)–output (local mechanical strains) relationship, and stiffness identification is conducted iteratively via correlating the calculated strain responses with measured data. The nonlinear model of the joint is reproduced with a number of linear local models in different deformation scenarios that are corresponding to different temperature ranges. The stiffness parameters can be identified using a multimodel approach. Numerical results show that the method is effective and reliable to identify the nonlinear connection stiffness of the ‘Que-Ti’ accurately with the temperature change even with 10% noise in measurements. The monitoring data from a long-term monitoring system installed in a typical heritage Tibetan building is used to further verify the method. The experimental results show that the identified stiffness by the proposed method with nonlinear connection stiffness model can get better results than that obtained from the linear connection stiffness model.

Мобильные диссипативные бризеры в цепочке нелинейных осцилляторов

In a numerical experiment, it was found that in a chain of Raleigh oscillators with a nonlinear connection between them, at least three localized stationary breathers modes may exist: non-mobile, "slow" and "fast" dissipative breathers. The dynamics of the chain elements depends on the ratio of characteristic time scales of the system which determined by the frequency of oscillators and the rigidity of the connection between them. Considered system is multistable.

On the connections of sub-Finslerian geometry

A sub-Finslerian manifold is, roughly speaking, a manifold endowed with a Finsler type metric which is defined on a [Formula: see text]-dimensional smooth distribution only, not on the whole tangent manifold. Our purpose is to construct a generalized nonlinear connection for a sub-Finslerian manifold, called [Formula: see text]-connection by the Legendre transformation which characterizes normal extremals of a sub-Finsler structure as geodesics of this connection. We also wish to investigate some of its properties like normal, adapted, partial and metrical.

Jacobi Stability Analysis of the Chen System

In this paper, the research of the Jacobi stability of the Chen system is performed by using the KCC-theory. By associating a nonlinear connection and a Berwald connection, five geometrical invariants of the Chen system are obtained. The Jacobi stability of the Chen system at equilibrium points and a periodic orbit is investigated in terms of the eigenvalues of the deviation curvature tensor. The obtained results show that the origin is always Jacobi unstable, while the Jacobi stability of the other two nonzero equilibrium points depends on the values of the parameters. And a periodic orbit of the Chen system is proved to be also Jacobi unstable. Furthermore, Jacobi stability regions of the Chen system and the Lorenz system are compared. Finally, the dynamical behavior of the components of the deviation vector near the equilibrium points is also discussed.

Super Finsler connection of superparticle on two-dimensional curved spacetime

We analyze the Casalbuoni–Brink–Schwarz superparticle model on a 2-dimensional curved spacetime as a super Finsler metric defined on a (2,2)-dimensional supermanifold. We propose a nonlinear Finsler connection which preserves this Finsler metric and calculates it explicitly. The equations of motion of the superparticle are reconstructed in the form of auto-parallel equations expressed by the super nonlinear connection.

Bounce Cosmology in Generalized Modified Gravities

We investigate the bounce realization in the framework of generalized modified gravities arising from Finsler and Finsler-like geometries. In particular, a richer intrinsic geometrical structure is reflected in the appearance of extra degrees of freedom in the Friedmann equations that can drive the bounce. We examine various Finsler and Finsler-like constructions. In the cases of general very special relativity, as well as of Finsler-like gravity on the tangent bundle, we show that a bounce cannot easily be obtained. However, in the Finsler–Randers space, induced scalar anisotropy can fulfil bounce conditions, and bouncing solutions are easily obtained. Finally, for the general class of theories that include a nonlinear connection, a new scalar field is induced, leading to a scalar–tensor structure that can easily drive a bounce. These features reveal the capabilities of Finsler and Finsler-like geometries.