Bifurcations, ultimate boundedness and singular orbits in a unified hyperchaotic Lorenz-type system

Abstract

In this note, by using the theory of bifurcation and Lyapunov function, one performs a qualitative analysis on a novel four-dimensional unified hyperchaotic Lorenz-type system (UHLTS), including stability, pitchfork bifurcation, Hopf bifurcation, singularly degenerate heteroclinic cycle, ultimate bound estimation, global exponential attractive set, heteroclinic orbit and so on. Numerical simulations not only are consistent with the results of theoretical analysis, but also illustrate singularly degenerate heteroclinic cycles with distinct geometrical structures and nearby hyperchaotic attractors in the case of small b > 0, i.e. conjugate hyperchaotic Lorenz-type attractors (CHCLTA) and nearby a short-duration transient of singularly degenerate heteroclinic cycles approaching infinity or singularly degenerate heteroclinic cycles consisting of normally hyperbolic saddle-foci and stable node-foci, etc. In particular, by a linear scaling, a possibly new forming mechanism behind the creation of well-known hyperchaotic attractor with (a 1 , a 2 , c, d, e, f, b, p, q) = (−12, 12, 23, −1, −1, 1, 2.1, −6, −0.2), consisting of occurrence of degenerate pitchfork bifurcation at Sz, the change in the stability index of the saddle at the origin as b crosses the null value, explosion of normally hyperbolic stable node-foci, collapse of singularly degenerate heteroclinic cycles consisting of normally hyperbolic saddle-foci or saddle-nodes, and stable node-foci, is revealed. The findings and results of this paper may provide theoretical support in some future applications, since they improve and complement the known ones.

Maximum likelihood least squares‐based iterative methods for output‐error bilinear‐parameter models with colored noises

“…Compared with the overparameterization model identification methods, the parameter separation technique introduced in this article can be applied to identify large-scale systems through the effective reduction of the number of parameters to be identified, thereby greatly reducing the computational cost and simplifying the identification procedure. The proposed methods in this article can be used for modeling and prediction and can be extended to study the parameter estimation problems of different systems with colored noises [47][48][49][50][51][52] and can be applied to other literatures [53][54][55] such as signal modeling and communication systems, engineering application systems, and so on.…”

The modified extended Kalman filter based recursive estimation for Wiener nonlinear systems with process noise and measurement noise

“…The basic idea in this article can be combined the mathematical tools 41,42 such as the data filtering technique, 43,44 the particle filter, 45,46 and the iterative methods [47][48][49] to study the identification of other linear stochastic systems, 50 bilinear stochastic systems, 51,52 and nonlinear stochastic systems [53][54][55][56][57] and can be applied to other engineering areas. [58][59][60][61] Theorem 3. For the Wiener nonlinear system in (1 )- (2 ) and the MEKF-RLS algorithm in (19 )- (26 ) and (51 )- (56 ), suppose that ( A1), ( A2), and ( A3) hold and there exist an integer N and positive constants 1 and 2 irrelevant to t such that for t ⩾ N, the following persistent excitation condition holds,…”

“…Equations ()‐() and ()‐() form the MEKF‐RLS algorithm for estimating the parameters of the Wiener system in (). The basic idea in this article can be combined the mathematical tools 41,42 such as the data filtering technique, 43,44 the particle filter, 45,46 and the iterative methods 47‐49 to study the identification of other linear stochastic systems, 50 bilinear stochastic systems, 51,52 and nonlinear stochastic systems 53‐57 and can be applied to other engineering areas 58‐61 …”

Auxiliary model multiinnovation stochastic gradient parameter estimation methods for nonlinear sandwich systems

“…Among the nonlinear block-oriented systems, the Wiener and Hammerstein structures are the most representative and were widely discussed in the literatures using some mathematical processing methods. 7,8 For the reason that the block-oriented nonlinear models can reflect real industrial nonlinear processes by simple cascade structures, they are widely used in aerospace systems, power systems, and so on. 9 Because the block-oriented nonlinear systems separate the nonlinear dynamical characteristic into the static nonlinearities and linear time-invariant dynamics, it is quite simple to understand and easy to be employed in industrial processes.…”

“…There are many kinds of structures for the nonlinear block‐oriented systems, which are composed of nonlinear static blocks and linear dynamical blocks to describe the complicated characteristics of industrial processes. Among the nonlinear block‐oriented systems, the Wiener and Hammerstein structures are the most representative and were widely discussed in the literatures using some mathematical processing methods 7,8 . For the reason that the block‐oriented nonlinear models can reflect real industrial nonlinear processes by simple cascade structures, they are widely used in aerospace systems, power systems, and so on 9 .…”