Modified slow-fast analysis method for slow-fast dynamical systems with two scales in frequency domain

Zhang, Zhengdi; Zhang-Yao, Chen; Bi, Qinsheng

doi:10.1016/j.taml.2019.05.010

Cited by 9 publications

(9 citation statements)

References 13 publications

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“…Furthermore, only two relatively simple types of bifurcations, i.e., the fold and Hopf bifurcations, are considered for the transitions between the quiescent states and spiking states [17]. When more equilibrium branches coexist, the limit cycles, including the cycles with relatively large amplitude near the homo-clinic orbits and the cycles bifurcated via Hopf bifurcations may interact with each other to form fold limit cycle bifurcations [18,19], which may lead more complicated bursting behaviors.…”

Section: Introductionmentioning

confidence: 99%

On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified Van der Pol–Duffing system with slow-varying periodic excitation

Zhang

Han

et al. 2022

Nonlinear Dyn

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The main purpose of the paper is to reveal the mechanism of certain special phenomena in bursting oscillations such as the sudden increase of the spiking amplitude. When multiple equilibrium points coexist in a dynamical system, several types of stable attractors via different bifurcations from these points may be observed with the variation of parameters, which may interact with each other to form other types of bifurcations. Here we take the modified van der Pol-Duffing system as an example, in which periodic parametric excitation is introduced. When the exciting frequency is far less than the natural frequency, bursting oscillations may appear. By regarding the exciting term as a slow-varying parameter, the number of the equilibrium branches in the fast generalized autonomous subsystem varies from one to five with the variation of the slowvarying parameter, on which different types of bifurcations, such as Hopf and pitch fork bifurcations, can be observed. The limit cycles, including the cycles via Hopf bifurcations and the cycles near the homo-clinic orbit may interact with each other to form the fold limit cycle bifurcations. With the increase of the exciting amplitude, different stable attractors and bifurcations of the generalized autonomous fast subsystem involve the full system, leading to different types of bursting oscillations. Fold limit cycle bifurcations may cause the sudden change of the spiking amplitude, since at the bifurcation points, the trajectory may oscillate according to different stable limit cycles with obviously dif-

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Section: Introductionmentioning

confidence: 99%

On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified Van der Pol–Duffing system with slow-varying periodic excitation

Zhang

Han

et al. 2022

Nonlinear Dyn

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“…Although the rise-dimension method seems to increase the difficulty in the analysis of (3), it should be pointed out that only the slow variable w appears in the fast subsystem (4), indicating that the dynamics structures of the fast subsystem only can be affected by w. As a consequence, the classical slow-fast decomposition method can be directly employed to explain the generation mechanism of slow-fast dynamics behaviors in (3) just via regarding the slow variable w as a bifurcation parameter in (4). Meanwhile, the rise-dimension method also demonstrates that the generalized slow-fast analysis method proposed in [42] is feasible mathematically and logically.…”

Section: Mathematical Modelmentioning

confidence: 93%

“…To name but a few, canards and chaotic bursting were reproduced in memristor-based Chua's circuit [40]; Marszalek and Trzaska analyzed mixed-mode oscillations and relaxations in slow-fast modified Chua's circuit models with autonomous vector fields [41]. Particularly, those modified Chua's circuits with low-frequency excitations are also employed to investigate various bursting oscillations and the generation mechanisms [24][25][26]42].…”

Section: Mathematical Modelmentioning

confidence: 99%

Slow–Fast Dynamics Behaviors under the Comprehensive Effect of Rest Spike Bistability and Timescale Difference in a Filippov Slow–Fast Modified Chua’s Circuit Model

Chen

et al. 2022

Mathematics

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Since the famous slow–fast dynamical system referred to as the Hodgkin–Huxley model was proposed to describe the threshold behaviors of neuronal axons, the study of various slow–fast dynamical behaviors and their generation mechanisms has remained a popular topic in modern nonlinear science. The primary purpose of this paper is to introduce a novel transition route induced by the comprehensive effect of special rest spike bistability and timescale difference rather than a common bifurcation via a modified Chua’s circuit model with an external low-frequency excitation. In this paper, we attempt to explain the dynamical mechanism behind this novel transition route through quantitative calculations and qualitative analyses of the nonsmooth dynamics on the discontinuity boundary. Our work shows that the whole system responses may tend to be various and complicated when this transition route is triggered, exhibiting rich slow–fast dynamics behaviors even with a very slight change in excitation frequency, which is described well by using Poincaré maps in numerical simulations.

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“…These may lead to unstable oscillations in the power system over a wide frequency range. To accommodate this change, based on singular perturbation and stability region theory, a timescale decomposition method for power system stability analysis is proposed in [47], [51], [53]. This establishes a link between the traditional large-and small-signal stability analysis, and is applicable to the analysis of multi-timescale systems.…”

Section: ) Classificationmentioning

confidence: 99%

“…With the increasing size and capacity of MGs, it is necessary to explore the multi-timescale characteristics of MGs to gain insight into the dynamic characteristics of the system as well as the instability mechanism. The decomposition of the stability analysis in a timescale can be used to analyze the stability of multi-timescale systems such as MG systems [45], [47], [51] [53]. Based on singular perturbation and stability domain theory, the method can be used to capture instabilities caused by interactions between fast and slow dynamics, avoiding wrong conclusions from transient and quasi-steady-state analyses [47], [51].…”

Section: ⅰ Introductionmentioning

confidence: 99%

Multi-Timescale Modeling and Dynamic Stability Analysis for Sustainable Microgrids: State-of-the-Art and Perspectives

Zhang,

Han,

Liu

et al. 2024

Prot. Control Mod. Power Syst.

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The increasing trend for integrating renewable energy sources into the grid to achieve a cleaner energy system is one of the main reasons for the development of sustainable microgrid (MG) technologies. As typical power-electronized power systems, MGs make extensive use of power electronics converters, which are highly controllable and flexible but lead to a profound impact on the dynamic performance of the whole system. Compared with traditional large-capacity power systems, MGs are less resistant to perturbations, and various dynamic variables are coupled with each other on multiple timescales, resulting in a more complex system in-stability mechanism. To meet the technical and economic challenges, such as active and reactive power-sharing, volt-age, and frequency deviations, and imbalances between power supply and demand, the concept of hierarchical control has been introduced into MGs, allowing systems to control and manage the high capacity of renewable energy sources and loads. However, as the capacity and scale of the MG system increase, along with a multi-timescale control loop design, the multi-timescale interactions in the system may become more significant, posing a serious threat to its safe and stable operation. To investigate the multi-timescale behaviors and instability mechanisms under dynamic inter-actions for AC MGs, existing coordinated control strategies are discussed, and the dynamic stability of the system is defined and classified in this paper. Then, the modeling and assessment methods forthe stability analysis of multi-timescale systems are also summarized. Finally, an outlook and discussion of future research directions for AC MGs are also presented.

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Modified slow-fast analysis method for slow-fast dynamical systems with two scales in frequency domain

Cited by 9 publications

References 13 publications

On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified Van der Pol–Duffing system with slow-varying periodic excitation

On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified Van der Pol–Duffing system with slow-varying periodic excitation

Slow–Fast Dynamics Behaviors under the Comprehensive Effect of Rest Spike Bistability and Timescale Difference in a Filippov Slow–Fast Modified Chua’s Circuit Model

Multi-Timescale Modeling and Dynamic Stability Analysis for Sustainable Microgrids: State-of-the-Art and Perspectives

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