scholarly journals BIFURCATIONS OF SNAP-BACK REPELLERS WITH APPLICATION TO BORDER-COLLISION BIFURCATIONS

2010 ◽  
Vol 20 (02) ◽  
pp. 479-489 ◽  
Author(s):  
PAUL GLENDINNING
Keyword(s):  
Dynamical Systems ◽  
Normal Form ◽  
Bifurcation Theory ◽  
Hybrid Dynamical Systems ◽  
Border Collision Bifurcations ◽  
The Creation ◽  
Noninvertible Maps ◽  
Homoclinic Tangencies ◽  
Theoretical Results ◽  
Infinite Sequences

The bifurcation theory of snap-back repellers in hybrid dynamical systems is developed. Infinite sequences of bifurcations are shown to arise due to the creation of snap-back repellers in noninvertible maps. These are analogous to the cascades of bifurcations known to occur close to homoclinic tangencies for diffeomorphisms. The theoretical results are illustrated with reference to bifurcations in the normal form for border-collision bifurcations.

$\mathcal{KL}_*$ -stability for a class of hybrid dynamical systems

2017 ◽  
Vol 82 (5) ◽  
pp. 1043-1060 ◽  
Author(s):  
Bin Liu ◽  
Hai Huyen Heidi Dam ◽  
Kok Lay Teo ◽  
David John Hill
Keyword(s):  
Dynamical Systems ◽  
Dwell Time ◽  
Average Dwell Time ◽  
Hybrid Dynamical Systems ◽  
Uniform Asymptotic Stability ◽  
Time Condition ◽  
Event Time ◽  
Class Function ◽  
The Stability ◽  
Theoretical Results

Abstract This article studies $\mathcal{KL}_*$-stability (the stability expressed by $\mathcal{KL}_*$-class function) for a class of hybrid dynamical systems (HDS). The notions of $\mathcal{KL}_{*}\mathcal{K}_{*}$-property and $\mathcal{KL}_{*}$-stability are proposed for HDS with respect to the hybrid-event-time. The $\mathcal{KL}_{*}$-stability, which is based on $\mathcal{K}$ or $\mathcal{L}$ property of the continuous flow, the discrete jump, and the event in an HDS, extends the $\mathcal{KLL}$-stability and the event-stability reported in the literature for HDS. The relationships between $\mathcal{KL}_{*}\mathcal{K}_{*}$-property and $\mathcal{KL}_{*}$-stability are established via introducing the hybrid dwell-time condition (HDT). The HDT generalizes the average dwell-time condition in the literature. For an HDS with $\mathcal{KL}_{*}\mathcal{K}_{*}$-property consisting of stabilizing $\mathcal{L}$-property and destabilizing $\mathcal{K}$-property, it is shown that there exists a common HDT under which the HDS will achieve $\mathcal{KL}_{*}$-stability. Thus HDT may help to derive some easily tested conditions for HDS to achieve uniform asymptotic stability. Moreover, a criterion of $\mathcal{KL}_{*}$-stability is derived by using the multiple Lyapunov-like functions. Examples are given to illustrate the obtained theoretical results.

scholarly journals The Salted Kalman Filter: Kalman filtering on hybrid dynamical systems

Automatica ◽  
2021 ◽  
Vol 131 ◽  
pp. 109752
Author(s):  
Nathan J. Kong ◽  
J. Joe Payne ◽  
George Council ◽  
Aaron M. Johnson
Keyword(s):  
Kalman Filter ◽  
Dynamical Systems ◽  
Kalman Filtering ◽  
Hybrid Dynamical Systems

Symmetry-breaking bifurcations in resonant surface waves

1989 ◽  
Vol 199 ◽  
pp. 495-518 ◽  
Author(s):  
Z. C. Feng ◽  
P. R. Sethna
Keyword(s):  
Normal Form ◽  
Symmetry Breaking ◽  
Surface Waves ◽  
Bifurcation Analysis ◽  
Internal Resonance ◽  
Travelling Waves ◽  
Chaotic Behaviour ◽  
Derived Set ◽  
Parameter Values ◽  
Theoretical Results

Surface waves in a nearly square container subjected to vertical oscillations are studied. The theoretical results are based on the analysis of a derived set of normal form equations, which represent perturbations of systems with 1:1 internal resonance and with D4 symmetry. Bifurcation analysis of these equations shows that the system is capable of periodic and quasi-periodic standing as well as travelling waves. The analysis also identifies parameter values at which chaotic behaviour is to be expected. The theoretical results are verified with the aid of some experiments.

scholarly journals Identification of a Class of Hybrid Dynamical Systems

2020 ◽  
Vol 53 (2) ◽  
pp. 875-882
Author(s):  
Stefano Massaroli ◽  
Federico Califano ◽  
Angela Faragasso ◽  
Mattia Risiglione ◽  
Atsushi Yamashita ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Hybrid Dynamical Systems

scholarly journals Stability analysis and controller synthesis for hybrid dynamical systems

2010 ◽  
Vol 368 (1930) ◽  
pp. 4937-4960 ◽  
Author(s):  
W. P. M. H. Heemels ◽  
B. De Schutter ◽  
J. Lunze ◽  
M. Lazar
Keyword(s):  
Dynamical Systems ◽  
Mathematical Models ◽  
Hybrid Systems ◽  
Discrete Event ◽  
Research Area ◽  
Hybrid Dynamical Systems ◽  
Technological Systems ◽  
Analysis And Synthesis ◽  
Control Community ◽  
The One

Wherever continuous and discrete dynamics interact, hybrid systems arise. This is especially the case in many technological systems in which logic decision-making and embedded control actions are combined with continuous physical processes. Also for many mechanical, biological, electrical and economical systems the use of hybrid models is essential to adequately describe their behaviour. To capture the evolution of these systems, mathematical models are needed that combine in one way or another the dynamics of the continuous parts of the system with the dynamics of the logic and discrete parts. These mathematical models come in all kinds of variations, but basically consist of some form of differential or difference equations on the one hand and automata or other discrete-event models on the other hand. The collection of analysis and synthesis techniques based on these models forms the research area of hybrid systems theory, which plays an important role in the multi-disciplinary design of many technological systems that surround us. This paper presents an overview from the perspective of the control community on modelling, analysis and control design for hybrid dynamical systems and surveys the major research lines in this appealing and lively research area.

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