Complex dynamics of a three-dimensional continuous-time autonomous system

Chaotic systems with a curve of equilibria have attracted considerable interest in theoretical researches and engineering applications because they are categorized as systems with hidden attractors. In this paper, we introduce a new three-dimensional autonomous system with cubic equilibrium. Fundamental dynamical properties and complex dynamics of the system have been investigated. Of particular interest is the coexistence of chaotic attractors in the proposed system. Furthermore, we have designed and implemented an electronic circuit to verify the feasibility of such a system with cubic equilibrium.

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2 × 2-SCROLL ATTRACTORS GENERATED IN A THREE-DIMENSIONAL SMOOTH AUTONOMOUS SYSTEM

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127407019688 ◽

2007 ◽

Vol 17 (11) ◽

pp. 4153-4157 ◽

Cited By ~ 8

Author(s):

WENBO LIU ◽

WALLACE K. S. TANG ◽

GUANRONG CHEN

Keyword(s):

Numerical Simulations ◽

Continuous Time ◽

Autonomous System ◽

Electronic Circuit ◽

Three Dimensional ◽

Chaotic Attractors ◽

First Time

In this letter, a three-dimensional continuous-time smooth autonomous system with quadratic nonlinear terms is proposed for generating multiscroll chaotic attractors. Observation of 2 × 2-scroll attractors generated from this kind of system is reported for the first time. The result is confirmed by both numerical simulations and electronic circuit experiments.

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A CHAOTIC SYSTEM WITH ONE SADDLE AND TWO STABLE NODE-FOCI

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127408021063 ◽

2008 ◽

Vol 18 (05) ◽

pp. 1393-1414 ◽

Cited By ~ 118

Author(s):

QIGUI YANG ◽

GUANRONG CHEN

Keyword(s):

Chaotic System ◽

Chaotic Attractor ◽

Autonomous System ◽

Lorenz System ◽

Complex Dynamics ◽

Three Dimensional ◽

Type Systems ◽

Stable Node ◽

Compound Structure ◽

Chen System

This paper reports the finding of a chaotic system with one saddle and two stable node-foci in a simple three-dimensional (3D) autonomous system. The system connects the original Lorenz system and the original Chen system and represents a transition from one to the other. The algebraical form of the chaotic attractor is very similar to the Lorenz-type systems but they are different and, in fact, nonequivalent in topological structures. Of particular interest is the fact that the chaotic system has a chaotic attractor, one saddle and two stable node-foci. To further understand the complex dynamics of the system, some basic properties such as Lyapunov exponents, bifurcations, routes to chaos, periodic windows, possible chaotic and periodic-window parameter regions, and the compound structure of the system are analyzed and demonstrated with careful numerical simulations.

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A NEW HYPERCHAOTIC SYSTEM AND ITS CIRCUIT IMPLEMENTATION

International Journal of Bifurcation and Chaos ◽

10.1142/s021812741002640x ◽

2010 ◽

Vol 20 (04) ◽

pp. 1201-1208 ◽

Cited By ~ 9

Author(s):

MINGHUA LIU ◽

JIUCHAO FENG ◽

CHI K. TSE

Keyword(s):

Nonlinear Control ◽

Bifurcation Diagram ◽

Continuous Time ◽

Control Parameter ◽

Electronic Circuit ◽

Three Dimensional ◽

Hyperchaotic System ◽

Circuit Implementation ◽

Time Autonomous ◽

New Hyperchaotic System

A four-dimensional continuous-time autonomous hyperchaotic system is proposed in this letter. This system is constructed by incorporating a nonlinear control to a three-dimensional continuous-time autonomous chaotic system. The hyperchaotic system is analyzed by studying the spectrum of Lyapunov exponents and the corresponding bifurcation diagram. The system exhibits chaotic, periodic, hyperchaotic behaviors for different values of a selected control parameter. Also, a simple electronic circuit is designed and implemented. Simulations and experimental observations verify the analytical results.

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A New Feigenbaum-Like Chaotic 3D System

Discrete Dynamics in Nature and Society ◽

10.1155/2014/328143 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Huitao Zhao ◽

Yiping Lin ◽

Yunxian Dai

Keyword(s):

Lyapunov Exponent ◽

Autonomous System ◽

Complex Dynamics ◽

Three Dimensional ◽

Period Doubling ◽

Fractional Dimension ◽

Largest Lyapunov Exponent ◽

System A ◽

Route To Chaos ◽

Doubling Sequence

Based on Sprott N system, a new three-dimensional autonomous system is reported. It is demonstrated to be chaotic in the sense of having positive largest Lyapunov exponent and fractional dimension. To further understand the complex dynamics of the system, some basic properties such as Lyapunov exponents, bifurcation diagram, Poincaré mapping, and period-doubling route to chaos are analyzed with careful numerical simulations. The obtained results also show that the period-doubling sequence of bifurcations leads to a Feigenbaum-like strange attractor.

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Managing Self-Replicating Innovative Goods

Management Science ◽

10.1287/mnsc.2020.3936 ◽

2021 ◽

Author(s):

Bin Hu ◽

Zhankun Sun

Keyword(s):

Innovation Diffusion ◽

Operations Management ◽

Continuous Time ◽

Three Dimensional ◽

Production Capacity ◽

Marketing Strategies ◽

Control Framework ◽

Time Optimal ◽

Inventory Decision ◽

Self Replication

Inspired by self-replicating three-dimensional printers and innovative agricultural and husbandry goods, we study optimal production and sales policies for a manufacturer of self-replicating innovative goods with a focus on the unique “keep-or-sell” trade-off—namely, whether a newly produced unit should be sold to satisfy demand and stimulate future demand or added to inventory to increase production capacity. We adopt the continuous-time optimal control framework and marry a self-replication model on the production side to the canonical innovation-diffusion model on the demand side. By analyzing the model, we identify a condition that differentiates Strong and Weak Replicability regimes, wherein production and sales, respectively, take priority over the other and fully characterize their distinct optimal policies. These insights prove robust and helpful in several extensions, including backlogged demand, liquidity constraints, stochastic innovation diffusion, launch inventory decision, and exogenous demand. We also find that social marketing strategies are particularly well suited for self-replicating innovative goods under Strong Replication. This paper was accepted by Victor Martínez de Albéniz, operations management.

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Stability of L1 contractions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004115000158 ◽

2015 ◽

Vol 159 (1) ◽

pp. 23-46

Author(s):

LUIS BARREIRA ◽

CLAUDIA VALLS

Keyword(s):

Continuous Time ◽

Autonomous System ◽

Nonautonomous System ◽

Nonlinear Perturbations ◽

Variation Of Parameters ◽

Natural Development ◽

Linear And Nonlinear ◽

Variation Of Parameters Formula ◽

Type Of Stability

AbstractThe notion of an exponential contraction is only one among many possible rates of contraction of a nonautonomous system, while for an autonomous system all contractions are exponential. We consider the notion of an L1 contraction that includes exponential contractions as a very particular case and that is naturally adapted to the variation-of-parameters formula. Both for discrete and continuous time, we show that under very general assumptions the notion of an L1 contraction persists under sufficiently small linear and nonlinear perturbations, also maintaining the type of stability. As a natural development, we establish a version of the Grobman–Hartman theorem for nonlinear perturbations of an L1 contraction.

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