## Abstract

An iterative method (AIM) is one of the numerical method, which is easy to apply and very time convenient for solving nonlinear differential equations. However, if we want to work in a large interval, sometimes it may be difficult to apply AIM. Therefore, a multistage AIM named Multistage Modified Iterative Method (MMIM) is introduced in this article to work in a large computational interval. The applicability of MMIM for increasing the solution domain of the given problems is construed in this article. Some problems are solved numerically using MMIM, which provides a better result in the extended interval as compared to AIM. Comparison tables and some graphs are included to demonstrate the results.

## Abstract

We prove that given any closed 3-manifold *M*
^{3}, there is an A-flow *f*
^{t} on *M*
^{3} such that the non-wandering set *NW* (*f*
^{t}) consists of 2-dimensional non-orientable expanding attractor and trivial basic sets.

## Abstract

The article continues a series of works studying cylindrical transformations having discrete orbits (Besicovitch cascades). For any *γ* ∈ (0,1) and any *ɛ* > 0 we construct a Besicovitch cascade over some rotation with bounded partial quotients, and with a *γ*–Hölder function, such that the Hausdorff dimension of the set of points in the circle having discrete orbits is greater than 1 − *γ*− *ɛ*.

## Abstract

The purpose of this article is to review the author's results on the existence and structure of minimal sets and attractors of conformal foliations. Results on strong transversal equivalence of conformal foliations are also presented. Connections with works of other authors are indicated. Examples of conformal foliations with exceptional, exotic and regular minimal sets which are attractors are constructed.

## Abstract

We study behavior of the topological entropy as the function of parameters for two-parameter family of symmetric Lorenz maps *T*
_{c,ɛ}(*x*) = (−1 + *c|x|*
^{1−ɛ}) · *sgn*(*x*). This is the normal form for splitting the homoclinic loop in systems which have a saddle equilibrium with one-dimensional unstable manifold and zero saddle value. Due to L.P. Shilnikov results, such a bifurcation corresponds to the birth of Lorenz attractor (when the saddle value becomes positive). We indicate those regions in the bifurcation plane where the topological entropy depends monotonically on the parameter *c*, as well as those for which the monotonicity does not take place. Also, we indicate the corresponding bifurcations for the Lorenz attractors.

## Abstract

The rapid growth of fossil fuels and the aggravation of emission pollution can be improved by the green and diversified energy structure of road transport vehicles. In this article, Cui–Lawson model is first introduced to analyse the change trend of pure electric vehicle, hybrid electric vehicle, gas–fuel vehicle, fuel cell vehicle, biofuel vehicle and traditional vehicle in the next 50 years. Considering the climate, environment and economic development of each region, the future ownership of the above six models was studied by region. Moreover, the climate and topography affect the developing trend strongly. The ownership of traditional automobile is more than other kinds of vehicles until 35 and 40 years in Xibei and Dongbei regions for the cold temperature.

## Abstract

In the present paper we construct an example of 4-dimensional flows on 𝕊^{3} × 𝕊^{1} whose saddle periodic orbit has a wildly embedded 2-dimensional unstable manifold. We prove that such a property has every suspension under a non-trivial Pixton's diffeomorphism. Moreover we give a complete topological classification of these suspensions.

## Abstract

For a collection of infinite loaded graphs, random perturbations of special type are considered. It is shown that some known classes of these graphs are stable with respect to small random perturbations of this type, while the rest are not.

## Abstract

The behavior of a viscous incompressible fluid on a rotating sphere is described by the nonlinear barotropic vorticity equation (BVE). Conditions for the existence of a bounded set that attracts all BVE solutions are given. In addition, sufficient conditions are obtained for a BVE solution to be a global attractor. It is shown that, in contrast to the stationary forcing, the dimension of the global BVE attractor under quasiperiodic forcing is not limited from above by the generalized Grashof number.

## Abstract

This article highlights particular mixed-mode oscillations (MMO) based on canard explosion observed in a fractional-order Fitzhugh-Nagumo (FFHN) model. In order to rigorously analyze the dynamics of the FFHN model, a recently introduced mathematical notion, the Hopf-like bifurcation (HLB), which provides a precise definition for the change between a fixed point and an *S*−asymptotically *T*−periodic solution, is used. The existence of HLB in this FFHN model is proved and the appearance of MMO based on canard explosion in the neighborhoods of such HLB points are numerically investigated using a new algorithm: the global-local canard explosion search algorithm. This MMO is constituted of various patterns of solutions with an increasing number of small-amplitude oscillations when two key parameters of the FFHN model are varied simultaneously. On the basis of such numerical experiment, it is conjectured that chaos could occur in a two-dimensional fractional-order autonomous dynamical system, with the fractional-order close to one. Therefore, this very simple two-dimensional FFHN model, presents an incredible ability to mimic the complex dynamics of neurons.