Instability and controllability of linearly coupled oscillators: Eigenvalue analysis

Hu, Gang; Yang, Junzhong; Liu, Wenji

doi:10.1103/physreve.58.4440

Cited by 123 publications

(88 citation statements)

References 21 publications

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“…We show that, essentially, the problem can be reduced to analyzing the maximal eigenvalue of a correlated random matrix, as discussed in the previous section [46,47,48].…”

Section: A Theoretical Remarkmentioning

confidence: 99%

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Synchronization in networks with random interactions: Theory and applications

Feng

Jirsa²,

Ding³

2006

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Synchronization is an emergent property in networks of interacting dynamical elements. Here we review some recent results on synchronization in randomly coupled networks. Asymptotical behavior of random matrices is summarized and its impact on the synchronization of network dynamics is presented. Robert May's results on the stability of equilibrium points in linear dynamics are first extended to nonlinear systems where the synchronized dynamics can be periodic or chaotic and then to systems with time delayed coupling. Finally, applications of our results to neuroscience, in particular, networks of Hodgkin-Huxley neurons, are included.Behaviour and structures of interacting dynamical systems are complex. Most research on complex networks has been focused so far on networks with deterministic interactions. In this paper we study the behaviour of networks with random interactions. We extend Robert May's results on the stability of equilibrium points in linear dynamics to nonlinear systems where the synchronized dynamics can be periodic or chaotic and then to systems with time delayed coupling. Results are then applied to networks in Neuroscience.

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“…We show that, essentially, the problem can be reduced to analyzing the maximal eigenvalue of a correlated random matrix, as discussed in the previous section [46,47,48].…”

Section: A Theoretical Remarkmentioning

confidence: 99%

“…In the first example we consider the coupled differential equation systems with H(x) = x [47]. It is easy to see that DH is a M × M identity matrix.…”

Section: Examplesmentioning

confidence: 99%

Synchronization in networks with random interactions: Theory and applications

Feng

Jirsa²,

Ding³

2006

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…Obviously, the invariant ILDM is given by w(t)¼ u(t)-S(t)¼ 0. Therefore, the stability of the ILDM can be analyzed by dwðtÞ dt ¼ duðtÞ dt À dSðtÞ dt , which can be studied by the following linearized equation: 27,28 …”

Section: Stability Analysis Of the Ildmmentioning

confidence: 99%

“…[23][24][25][26] Some detailed analysis of chaos dynamics can be found in the literature. 27,28 Mathematically, a chaotic dynamics also exhibits dense periodic orbits and topologically mixing of its phase space open sets. 27,28 Chaos has been observed in a vast variety of realistic systems, including Belouzov-Zhabotinski reactions, nonlinear optics, Chua-Matsumoto circuit, Rayleigh-Benard convention, meteorology, population dynamics, psychology, economics, finance solar system, protein dynamics, 29 and heart and brain of living organisms.…”

Section: Introductionmentioning

confidence: 99%

Molecular nonlinear dynamics and protein thermal uncertainty quantification

Xia

Guo

2014

Chaos

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This work introduces molecular nonlinear dynamics (MND) as a new approach for describing protein folding and aggregation. By using a mode system, we show that the MND of disordered proteins is chaotic while that of folded proteins exhibits intrinsically low dimensional manifolds (ILDMs). The stability of ILDMs is found to strongly correlate with protein energies. We propose a novel method for protein thermal uncertainty quantification based on persistently invariant ILDMs. Extensive comparison with experimental data and the state-of-the-art methods in the field validate the proposed new method for protein B-factor prediction. V C 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4861202]Protein folding produces characteristic and functional three-dimensional structures from unfolded polypeptides or disordered coils. [1][2][3][4][5] The emergence of extraordinary complexity in the protein folding process poses astonishing challenges to theoretical modeling and computer simulations. 6,7 The present work introduces molecular nonlinear dynamics (MND), or molecular chaotic dynamics, as a theoretical framework for describing and analyzing protein folding. We represent the dynamics of macromolecular particles (i.e., atoms or coarse-grained superatoms) by a set of intrinsically chaotic oscillators. A geometry to topology mapping is employed to create driving and response relations among chaotic oscillators. We unveil the existence of intrinsically low dimensional manifolds (ILDMs) in the chaotic dynamics of folded proteins. Additionally, we reveal that the transition from disordered to ordered conformations in protein folding increases the transverse stability of the ILDM. Stated differently, protein folding reduces the chaoticity of the nonlinear dynamical system, and a folded protein has the best ability to tame chaos. Furthermore, we bring to light the connection between the ILDM stability and the thermodynamic stability, which enables us to quantify the disorderliness and relative energies of folded, misfolded, and unfolded protein states. Finally, we exploit chaos for protein uncertainty quantification and develop a robust chaotic algorithm for the prediction of Debye-Waller factors, or temperature factors, of protein structures.

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“…One of the ultimate goals in studying network synchronization is to understand how the network topology affects the synchronizability. In the simplest case (see below), the network synchronizability can be well measured by the eigenratio R [17,18,19,20], thus the above question degenerates to understanding the relationship between network structure and its eigenvalues. Since there are countless topological characters for networks, a natural question is addressed: what is the most important factor by which the synchroizability of the system is mainly determined?…”

mentioning

confidence: 99%

Better synchronizability predicted by crossed double cycle

2006

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In this brief report, we propose a network model named crossed double cycles, which are completely symmetrical and can be considered as the extensions of nearest-neighboring lattices. The synchronizability, measured by eigenratio R, can be sharply enhanced by adjusting the only parameter, crossed length m. The eigenratio R is shown very sensitive to the average distance L, and the smaller average distance will lead to better synchronizability. Furthermore, we find that, in a wide interval, the eigenratio R approximately obeys a power-law form as R ∼ L 1.5 .PACS numbers: 89.75, 05.45.Xt Synchronization is observed in a variety of natural, social, physical and biological systems [1], and has found applications in a variety of field including communications, optics, neural networks and geophysics [2,3,4,5,6,7]. The large networks of coupled dynamical systems that exhibit synchronized state are subjects of great interest. In the early stage, the corresponding studies are restricted to either the regular networks [8,9], or the random ones [10,11]. However, recent empirical studies have demonstrated that many real-life networks can not be treated as regular or random networks. The most important two of their common statistic characteristics are called small-world effect [12] and scale-free property [13]. Therefore, very recently, most of the studies about network synchronization focus on complex networks, and find that the networks of small-world effect and scalefree property may be easier to synchronize than regular lattices [14,15,16]. One of the ultimate goals in studying network synchronization is to understand how the network topology affects the synchronizability. In the simplest case (see below), the network synchronizability can be well measured by the eigenratio R [17,18,19,20], thus the above question degenerates to understanding the relationship between network structure and its eigenvalues. Since there are countless topological characters for networks, a natural question is addressed: what is the most important factor by which the synchroizability of the system is mainly determined? Some previous works indicated the average distance L[21] is one of the key factors. However, the consistent conclusion have not been achieved [19,22,23,24,25]. Another extensively studied one is the network heterogeneity, which can be measured by the variance of degree distribution or betweenness distribution [26,27]. Some detailed comparisons among various networks have been done, indicating the network synchronizability will be better with smaller heterogeneity [24,28,29]. However, a well-known counterexample is the regular networks with homogeneous structure while displaying very poor synchronizability. Because of the networks used for comparing in previous studies are of both varying average distances and degree variances, the strict and clear conclusions can not be achieved. In addition, some researchers deem that the more intrinsic ingredient leading to better synchronizability is the randomicity [30], that is to say, ...

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Instability and controllability of linearly coupled oscillators: Eigenvalue analysis

Cited by 123 publications

References 21 publications

Synchronization in networks with random interactions: Theory and applications

Synchronization in networks with random interactions: Theory and applications

Molecular nonlinear dynamics and protein thermal uncertainty quantification

Better synchronizability predicted by crossed double cycle

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