In this technical note, the Hopf bifurcation in Chen's system is studied. Some corresponding dynamics are also discussed briefly.
In this paper, we investigate the problem of identifying or modeling nonlinear dynamical systems undergoing periodic and period-like (recurrent) motions. For accurate identification of nonlinear dynamical systems, the persistent excitation condition is normally required to be satisfied. Firstly, by using localized radial basis function networks, a relationship between the recurrent trajectories and the persistence of excitation condition is established. Secondly, for a broad class of recurrent trajectories generated from nonlinear dynamical systems, a deterministic learning approach is presented which achieves locally-accurate identification of the underlying system dynamics in a local region along the recurrent trajectory. This study reveals that even for a random-like chaotic trajectory, which is extremely sensitive to initial conditions and is long-term unpredictable, the system dynamics of a nonlinear chaotic system can still be locally-accurate identified along the chaotic trajectory in a deterministic way. Numerical experiments on the Rossler system are included to demonstrate the effectiveness of the proposed approach.
In this paper we study the chaotification problem of polynomial continuous-time systems in a semiglobal setting. Our results are based on the computation of rational normal forms and time-delay anticontroller design. As examples, the Rössler system, some Sprott systems and the Lorenz system are considered.
When a transmission delay occurs in the interconnection of linearly coupled systems described by ordinary differential equations (LCODEs), both synchronization and the final synchronized state will vary. In this paper, mathematical analysis is presented on the synchronization phenomena of LCODEs with a single coupling delay. Criteria are derived for both local and global synchronization. It is known that in addition to the dynamical behaviors of the underlying uncoupled system and the coupling configuration, the coupling strength and the coupling delay also play key roles on the stability of synchronization. Both theoretical and numerical analysis indicate that under some conditions, if the coupling strength is large enough, the coupled system can be completely synchronized for any coupling delay. On the other hand, in some cases, the coupled system can be synchronized if the coupling delay is small enough.
The diversity–stability relationship in system dynamics had found significant implication to species evolution in the classical cascade and niche models of the ecosystem. In this paper, the local diversity–stability of the q-snapback network model, which is similar to the cascade and niche models, is investigated by analyzing the eigenvalue distribution near its equilibrium. It is found that the basic diversity–stability feature of the q-snapback network is similar to that of the random-graph network, and to that of the cascade and niche models when their different degree distributions are set to be homogeneous.
This paper investigates some chaotic properties via Furstenberg families generated by inverse limit dynamical systems. It is proved that the inverse limit dynamical system ( lim ⟵ ( X , f ) , σ f ) of a dynamical system ( X , f ) is ℱ -transitive (resp., ℱ -mixing, ( ℱ 1 , ℱ 2 ) -everywhere chaotic) if and only if the system ( ∩ n = 0 ∞ f n ( X ) , f | ∩ n = 0 ∞ f n ( X ) ) is ℱ -transitive (resp., ℱ -mixing, ( ℱ 1 , ℱ 2 ) -everywhere chaotic), where ℱ , ℱ 1 and ℱ 2 are Furstenberg families.
This paper proves that a binary operation ⋆ on [0, 1], ensuring that the binary operation ⋏ is a t-norm or ⋎ is a t-conorm, is a t-norm, where ⋏ and ⋎ are special convolution operations defined by ( f ⋏ g ) ( x ) = sup { f ( y ) ★ g ( z ) : y ▵ z = x } , ( f ⋎ g ) ( x ) = sup { f ( y ) ★ g ( z ) : y ▿ z = x } , for any f, g ∈ Map([0, 1], [0, 1]), where △ and ▽ are a continuous t-norm and a continuous t-conorm on [0, 1], answering negatively an open problem posed in [8]. Besides, some characteristics of t-norm and t-conorm are obtained in terms of the binary operations ⋏ and ⋎.
No abstract is provided for this article.
This paper extends the work on discovering fuzzy association rules with degrees of support and implication (ARsi). The effort is twofold: one is to discover ARsi with hierarchy so as to express more semantics due to the fact that hierarchical relationships usually exist among fuzzy sets associated with the attribute concerned; the other is to generate a “core” set of rules, namely the rule cover set, that are of more interest in a sense that all other rules could be derived by the cover set. Corresponding algorithms for ARsi with hierarchy and the cover set are proposed along with pruning strategies incorporated to improve the computational efficiency. Some data experiments are conducted as well to show the effectiveness of the approach.
No abstract is provided for this article.
This paper establishes topological (equi-)semiconjugacy and (equi-)conjugacy between induced non-autonomous set-valued systems and subshifts of finite type. First, some necessary and sufficient conditions are given for a non-autonomous discrete system to be topologically semiconjugate or conjugate to a subshift of finite type. Further, several sufficient conditions for it to be topologically equi-semiconjugate or equi-conjugate to a subshift of finite type are obtained. Consequently, estimations of topological entropy and several criteria of Li-Yorke chaos and distributional chaos in a sequence are derived. Second, the relationships of several related dynamical behaviors between the non-autonomous discrete system and its induced set-valued system are investigated. Based on these results, the paper furthermore establishes the topological (equi-)semiconjugacy and (equi-)conjugacy between induced set-valued systems and subshifts of finite type. Consequently, estimations of the topological entropy for the induced set-valued system are obtained, and several criteria of Li-Yorke chaos and distributional chaos in a sequence are established. Some of these results not only extend the existing related results for autonomous discrete systems to non-autonomous discrete systems, but also relax the assumptions of the counterparts in the literature. Two examples are finally provided for illustration.
