A Novel of New 7D Hyperchaotic System with Self-Excited Attractors and Its Hybrid Synchronization

In this study, a novel 7D hyperchaotic model is constructed from the 6D Lorenz model via the nonlinear feedback control technique. The proposed model has an only unstable origin point. Thus, it is categorized as a model with self-excited attractors. And it has seven equations which include 19 terms, four of which are quadratic nonlinearities. Various important features of the novel model are analyzed, including equilibria points, stability, and Lyapunov exponents. The numerical simulation shows that the new class exhibits dynamical behaviors such as chaotic and hyperchaotic. This paper also presents the hybrid synchronization for a novel model via Lyapunov stability theory.

Direct statistical simulation of low-order dynamosystems

In this paper, we investigate the effectiveness of direct statistical simulation (DSS) for two low-order models of dynamo action. The first model, which is a simple model of solar and stellar dynamo action, is third order and has cubic nonlinearities while the second has only quadratic nonlinearities and describes the interaction of convection and an aperiodically reversing magnetic field. We show how DSS can be used to solve for the statistics of these systems of equations both in the presence and the absence of stochastic terms, by truncating the cumulant hierarchy at either second or third order. We compare two different techniques for solving for the statistics: timestepping, which is able to locate only stable solutions of the equations for the statistics, and direct detection of the fixed points. We develop a complete methodology and symbolic package in Python for deriving the statistical equations governing the low-order dynamic systems in cumulant expansions. We demonstrate that although direct detection of the fixed points is efficient and accurate for DSS truncated at second order, the addition of higher order terms leads to the inclusion of many unstable fixed points that may be found by direct detection of the fixed point by iterative methods. In those cases, timestepping is a more robust protocol for finding meaningful solutions to DSS.

On the Poisson Stability to Study a Fourth-Order Dynamical System with Quadratic Nonlinearities

This article discusses the search procedure for Poincaré recurrences to classify solutions on an attractor of a fourth-order nonlinear dynamical system, using a previously developed high-precision numerical method. For the resulting limiting solution, the Lyapunov exponents are calculated, using the modified Benettin’s algorithm to study the stability of the found regime and confirm the type of attractor.

Linearizability of 2:−3 Resonant Systems with Quadratic Nonlinearities

In this paper, the linearizability of a 2:−3 resonant system with quadratic nonlinearities is studied. We provide a list of the conditions for this family of systems having a linearizable center. The conditions for linearizablity are obtained by computing the ideal generated by the linearizability quantities and its decomposition into associate primes. To successfully perform the calculations, we use an approach based on modular computations. The sufficiency of the obtained conditions is proven by several methods, mainly by the method of Darboux linearization.

Comparison of Reduction Methods for Finite Element Geometrically Nonlinear Beam Structures

The aim of this contribution is to present numerical comparisons of model-order reduction methods for geometrically nonlinear structures in the general framework of finite element (FE) procedures. Three different methods are compared: the implicit condensation and expansion (ICE), the quadratic manifold computed from modal derivatives (MD), and the direct normal form (DNF) procedure, the latter expressing the reduced dynamics in an invariant-based span of the phase space. The methods are first presented in order to underline their common points and differences, highlighting in particular that ICE and MD use reduction subspaces that are not invariant. A simple analytical example is then used in order to analyze how the different treatments of quadratic nonlinearities by the three methods can affect the predictions. Finally, three beam examples are used to emphasize the ability of the methods to handle curvature (on a curved beam), 1:1 internal resonance (on a clamped-clamped beam with two polarizations), and inertia nonlinearity (on a cantilever beam).