Mixed-mode oscillations and chaos from a simple second-order oscillator under weak periodic perturbation

Shimizu, Kazuo; Sekikawa, Munehisa; Inaba, Nobuyuki

doi:10.1016/j.physleta.2011.02.053

Cited by 47 publications

(18 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Many studies on fast-scale and slow-scale dynamics including its stability analysis have been reported [34][35][36]. Moreover, mixed-mode oscillation, which is observed in circuits, lasers, chemical, and biochemical oscillators, causes fast-scale and slow-scale dynamics too [37][38][39][40][41]. However, all these systems are smooth dynamical systems, whose qualitative properties are different from those of nonsmooth systems.…”

Section: Introductionmentioning

confidence: 99%

Relationship of fast‐scale and slow‐scale instabilities in switching circuit with multiple inputs

Asahara

Banerjee

Kousaka

2016

Circuit Theory & Apps

View full text Add to dashboard Cite

Power converter circuits, such as current-controlled or voltage-controlled converters and inverters often have multiple inputs in the controller. The multiple inputs cause high-frequency and low-frequency oscillations. In earlier studies, the characteristics of circuits in fast-scale and slow-scale dynamics have been investigated. However, in many cases, circuits with multiple inputs have three or more dimensional topology which makes detailed analysis difficult. In this paper, we analyze a simple interrupted electric circuit in order to understand essential characteristics of fast-scale and slow-scale dynamics. The advantage of this simple interrupted circuit is that it is possible to derive a 1-dimensional map, which facilitates rigorous studies. Based on the structure of the return map and the characteristic multiplier, we explain the characteristics of the system. We report the occurrence of pitchfork, period doubling, and border collision bifurcations in slow scale, and period doubling bifurcation in fast scale. We found that local bifurcation, which appears in fast-scale dynamics, does not significantly affect the global behavior of the system while instabilities in the slow-scale dynamics strongly affect the system behavior.

show abstract

Section: Introductionmentioning

confidence: 99%

Relationship of fast‐scale and slow‐scale instabilities in switching circuit with multiple inputs

Asahara

Banerjee

Kousaka

2016

Circuit Theory & Apps

View full text Add to dashboard Cite

show abstract

“…For example, Izhikevich [23] had given the classification of bursting by using bifurcation on the fast subsystem. Shimizu [24] discussed the mixed-mode oscillations of the BVP system with weak periodic perturbation. Chumakov [25] established a kinetic model of catalytic hydrogen oxidation and investigated the generated mechanism of bursting.…”

Section: Introductionmentioning

confidence: 99%

Slow-fast effect and generation mechanism of brusselator based on coordinate transformation

Hou

Shen

2016

Open Physics

View full text Add to dashboard Cite

Abstract:The Brusselator with different time scales, which behaves in the classical slow-fast effect, is investigated, and is characterized by the coupling of the quiescent and spiking states. In order to reveal the generation mechanism by using the slow-fast analysis method, the coordinate transformation is introduced into the classical Brusselator, so that the transformed system can be divided into the fast and slow subsystems. Furthermore, the stability condition and bifurcation phenomenon of the fast subsystem are analyzed, and the attraction domains of different equilibria are presented by theoretical analysis and numerical simulation respectively. Based on the transformed system, it could be found that the generation mechanism between the quiescent and spiking states is Fold bifurcation and change of the attraction domain of the fast subsystem. The results may also be helpful to the similar system with multiple time scales.

show abstract

“…Chaos theory has applications in several fields of science and engineering such as oscillators [32,33,34], lasers [35,36,37], chemical reactions [38,39,40], biology [41,42], ecology [43,44,45], neural networks [46,47,48,49], robotics [50,51,52], fuzzy logic [53,54], electrical circuits [55,56,57], etc.…”

Section: Introductionmentioning

confidence: 99%

“…Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 7/21/15 12:40 PM By substituting the parameter update law(34) into(33), we obtain the time-derivative of…”

mentioning

confidence: 99%

Hyperchaos, adaptive control and synchronization of a novel 5-D hyperchaotic system with three positive Lyapunov exponents and its SPICE implementation

Vaidyanathan¹,

Volos²,

Pham³

2014

Archives of Control Sciences

100

View full text Add to dashboard Cite

In this research work, a twelve-term novel 5-D hyperchaotic Lorenz system with three quadratic nonlinearities has been derived by adding a feedback control to a ten-term 4-D hyperchaotic Lorenz system with three quadratic nonlinearities. The 4-D hyperchaotic Lorenz system has the Lyapunov exponents L 1 = 0.3684, L 2 = 0.2174, L 3 = 0 and L 4 = −12.9513, and the Kaplan-Yorke dimension of this 4-D system is found as D KY = 3.0452. The 5-D novel hyperchaotic Lorenz system proposed in this work has the Lyapunov exponents L 1 = 0.4195, L 2 = 0.2430, L 3 = 0.0145, L 4 = 0 and L 5 = −13.0405, and the Kaplan-Yorke dimension of this 5-D system is found as D KY = 4.0159. Thus, the novel 5-D hyperchaotic Lorenz system has a maximal Lyapunov exponent (MLE), which is greater than the maximal Lyapunov exponent (MLE) of the 4-D hyperchaotic Lorenz system. The 5-D novel hyperchaotic Lorenz system has a unique equilibrium point at the origin, which is a saddle-point and hence unstable. Next, an adaptive controller is designed to stabilize the novel 5-D hyperchaotic Lorenz system with unknown system parameters. Moreover, an adaptive controller is designed to achieve global hyperchaos synchronization of the identical novel 5-D hyperchaotic Lorenz systems with unknown system parameters. Finally, an electronic circuit realization of the novel 5-D hyperchaotic Lorenz system using SPICE is described in detail to confirm the feasibility of the theoretical model.

show abstract

Mixed-mode oscillations and chaos from a simple second-order oscillator under weak periodic perturbation

Cited by 47 publications

References 17 publications

Relationship of fast‐scale and slow‐scale instabilities in switching circuit with multiple inputs

Relationship of fast‐scale and slow‐scale instabilities in switching circuit with multiple inputs

Slow-fast effect and generation mechanism of brusselator based on coordinate transformation

Hyperchaos, adaptive control and synchronization of a novel 5-D hyperchaotic system with three positive Lyapunov exponents and its SPICE implementation

Contact Info

Product

Resources

About