A simple and systematic approach is developed for modeling and adaptive control of an unknown (or uncertain) chaotic system, using only input–output data obtained from the underlying dynamical system. Gaussian fuzzy membership functions are used in conjunction with the least-squares principle for the modeling and control. Based on the fuzzy modeling, an adaptive controller is devised, which works through self-adjusting the means and variances of the Gaussian membership functions for adaptation. The design procedure is illustrated by using the chaotic Duffing oscillator as an example, on which simulation results demonstrate the effectiveness of the proposed methodology.
For a dynamical system described by a set of autonomous ordinary differential equations, an attractor can be a point, a periodic cycle, or even a strange attractor. Recently, a new chaotic system with only one stable equilibrium was described, which locally converges to the stable equilibrium but is globally chaotic. This paper further shows that for certain parameters, besides the point attractor and chaotic attractor, this system also has a coexisting stable limit cycle, demonstrating that this new system is truly complicated and interesting.
It has been demonstrated that a piecewise-linear system can generate chaos under suitable conditions. This paper proposes a novel method for simultaneously creating two symmetrical chaotic attractor––an upper-attractor and a lower-attractor––in a 3D linear autonomous system. Basically dynamical behaviors of this new chaotic system are further investigated. Especially, the chaos formation mechanism is explored by analyzing the structure of fixed points and the system trajectories.
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This paper introduces a novel bistable nonvolatile locally-active memristor model based on Chua's unfolding theorem to explore the influence of the local activity on the complexity of nonlinear systems. It is shown that the memristor has two asymptotically stable equilibrium points in its power-off plot with a negative memductance and a positive memductance, namely, it is nonvolatile. It is then shown that the fast switching between the two stable equilibrium points can be implemented by applying an appropriate voltage pulse. It is found that the memristor possesses four locally-active regions in its DC V-I plot, and a small signal equivalent circuit associated with a locally-active operating point of the memristor is designed using the small-signal analysis method, to explore its locally-active characteristics. Based on the small signal equivalent circuit, it is furthermore demonstrated that the nonvolatile locally-active memristor, when connected in parallel with a linear capacitor, oscillates periodically around a locally-active operating point via Hopf bifurcation. The dynamics of the memristor and its periodic oscillation are analyzed using the theory of local activity, pole-zero analysis of admittance functions, Hopf bifurcation and the edge of chaos. Finally, it is found that a chaotic oscillation evolved from the periodic oscillation appears by adding an energy-storage element (inductor) on the periodic oscillating circuit, which leads to the chaotic oscillation, coexisting attractors and various complex dynamical phenomena.
This article offers a brief review of the fundamental concepts of bifurcation and chaos in nonlinear dynamical and control systems. Both the time-domain and frequency-domain versions of the classical Hopf bifurcation theory are studied in detail. Generalized (or degenerate) Hopf bifurcation is also discussed. Theoretical analysis and potential applications of the bifurcation theory in power systems are introduced. Meanwhile, chaos and the route to chaos from period-doubling bifurcations are described. In particular, chaos and bifurcations in feedback control systems and adaptive control systems are addressed. Because a nonlinear control system is by nature a very complex nonautonomous dynamical system due essentially to the involving of the control input, understanding and utilizing the rich dynamics of nonlinear control systems have an important impact in the modern technology. It calls for new effort and endeavor devoted to this scientific and engineering challenge.
An observer-type of Kalman innovation filtering algorithm to find a practically implementable "best" Kalman filter, and such an algorithm based on the evolutionary programming (EP) optima-search technique, are proposed, for linear discrete-time systems with time-invariant unknown-but-hounded plant and noise uncertainties. The worst-case parameter set from the stochastic uncertain system represented by the interval form with respect to the implemented "best" filter is also found in this work for demonstrating the effectiveness of the proposed filtering scheme. The new EP-based algorithm utilizes the global optima-searching capability of EP to find the optimal Kalman filter and state estimates at every iteration, which include both the best possible worst case Interval and the optimal nominal trajectory of the Kalman filtering estimates of the system state vectors. Simulation results are included to show that the new algorithm yields more accurate estimates and is less conservative as compared with other related robust filtering schemes.
This paper is to report the observation that when the popular time-delayed feedback strategy is used for control purpose, it may actually create unwanted bifurcations. Hopf bifurcation created by delayed feedback control is the main concern of this article, but some other types of bifurcations are also observed to exist in such delayed-feedback control systems. The observations are illustrated by computer simulations.
In this paper, some characterizations about orbit invariants, p-scrambled points and scrambled sets are obtained.Applying these results solves a conjecture and two problems given in [X.Fu, Y. You,
In this paper, the dynamics of elementary cellular automata rule 42 is investigated in the bi-infinite symbolic sequence space. Rule 42, a member of Wolfram’s class II which was said to be simply as periodic before, actually defines a chaotic global attractor; that is, rule 42 is topologically mixing on its global attractor and possesses the positive topological entropy. Therefore, rule 42 is chaotic in the sense of both Li-Yorke and Devaney. Meanwhile, the characteristic function and the basin tree diagram of rule 42 are explored for some finite length of binary strings, which reveal its Bernoulli characteristics. The method presented in this work is also applicable to studying the dynamics of other rules, especially the 112 Bernoulli-shift rules of the elementary cellular automata.
This study proposes a Verilog HDL based design for a high efficiency multi-stage hierarchical memory system with a double data rate synchronous dynamic random access memory (DDR SDRAM) simulation model as the main memory. In many mainstream modern computing systems, memory access speed and latency are major bottlenecks that limit master input of processor performance. These problems have become more pronounced as processor computing power increases while memory speeds fail to keep pace. At the same time, also hoping that the main memory can be a DDR that is close to reality. To address these challenges, the proposed design introduces several key innovations. The overall system adopts a multi-stage hierarchical structure, which significantly reduces delays caused by the speed gap between the input and main memory to the greatest extent. Additionally, optimized memory control and scheduling strategies are employed through advanced timing control and command scheduling. Also, a DDR simulation model is integrated as the main memory, offering a realistic representation of data flow and access patterns. This model helps to accurately predict and validate memory control strategies under various real-world conditions. Through complete design and integration of these blocks, the system's performance is significantly enhanced, demonstrating improved efficiency in memory management and data transmission. This design lays the foundation for the future development of memory systems and provides a promising solution for the next generation of high efficiency computing environments.
