^{1}, Xing Zheng

^{2,3}and Meng Zhan

^{2,a)}

### Abstract

Oscillation death (also called amplitude death), a phenomenon of coupling induced stabilization of an unstable equilibrium, is studied for an *arbitrary* symmetric complex network with delay-coupled oscillators, and the critical conditions for its linear stability are explicitly obtained. All cases including one oscillator, a pair of oscillators, regular oscillatornetworks, and complex oscillatornetworks with delay feedback coupling, can be treated in a unified form. For an arbitrary symmetric network, we find that the corresponding smallest eigenvalue of the Laplacian *λ* _{ N } (0 >*λ* _{ N } ≥ –1) completely determines the death island, and as *λ* _{ N } is located within the insensitive parameter region for nearly all complex networks,*the death island keeps nearly the largest and does not sensitively depend on the complex network structures*. This insensitivity effect has been tested for many typical complex networks including Watts-Strogatz (WS) and Newman-Watts (NW) small world networks, general scale-free (SF) networks, Erdos-Renyi (ER) random networks, geographical networks, and networks with community structures and is expected to be helpful for our understanding of dynamics on complex networks.

Collective dynamical behaviors on complex networks have become the focus of intensive research recently in nonlinear sciences. The interplay between structure and dynamics is of fundamental importance. Clearly, both the network structure (for how the network nodes are connected) and local dynamics (for what the dynamical form is on the network nodes) could play crucial roles in determining the global dynamics and exhibiting the global function of the systems. It has been well discovered that the synchronization processes on network sensitively depend on the network structure. Namely, with the change of the network topology, the system synchronization ability will tremendously change accordingly. As a result, we may call it sensitivity effect of synchronization on networks. In this work, we studied another fairly less studied collective behavior, delay-induced oscillation death, and found that quite oppositely the oscillation death DOES NOT sensitively depend on the network structure. For nearly all different types of complex networks, the death island keeps the largest, and the discrepancy for its occurrence in the parameter plane is almost negligible. In this respect, this counter-intuitive feature may be termed as insensitivity effect of delay-induced oscillation death on complex networks. We believe that this is a novel type of generic behavior for collective dynamics on networks, which has never been reported before.

This work was supported by the Outstanding Oversea Scholar Foundation of Chinese Academy of Sciences (Bairenjihua) and the National Natural Science Foundation of China under Grant No. 11075202. We thank two anonymous reviewers for their very helpful suggestions.

I. INTRODUCTION

II. CONDITIONS FOR DELAY-INDUCED DEATH IN COMPLEX OSCILLATORNETWORKS

III. SOME EXAMPLES

A. One delayed-feedback oscillator

B. A pair of delay-coupled oscillators

C. N delay-coupled oscillators in regular networks

D. N delay-coupled oscillators in complex networks

IV. CONCLUSION AND DISCUSSIONS

### Key Topics

- Oscillators
- 59.0
- Networks
- 45.0
- Network topology
- 22.0
- Eigenvalues
- 17.0
- Coupled oscillators
- 10.0

## Figures

The case for one delayed-feedback chaotic Rössler oscillator showing four death islands in the (*τ*, *K*) parameter space. The island boundary curves are determined by *τ* _{ a }(*m*, *k*) and *τ* _{ b }(*m*, *K*) in Eq. (11). *m* = 0, 1, 2, and 3. The other two curves *τ* _{ c }(*m*, *K*) and *τ* _{ d }(*m*, *K*) are unnecessary. The insert is the enlargement of the smallest island (*m* = 3). The number of islands has been well predicted by Eq. (14). The numerical result represented by the open circles is in good agreement with the theoretic prediction.

The case for one delayed-feedback chaotic Rössler oscillator showing four death islands in the (*τ*, *K*) parameter space. The island boundary curves are determined by *τ* _{ a }(*m*, *k*) and *τ* _{ b }(*m*, *K*) in Eq. (11). *m* = 0, 1, 2, and 3. The other two curves *τ* _{ c }(*m*, *K*) and *τ* _{ d }(*m*, *K*) are unnecessary. The insert is the enlargement of the smallest island (*m* = 3). The number of islands has been well predicted by Eq. (14). The numerical result represented by the open circles is in good agreement with the theoretic prediction.

The death island of a pair of delay-coupled chaotic Rössler oscillators determined by the two critical curves *τ* _{1}(0, *k*) and *τ* _{2}(0, *K*) in Eq. (16). Now, *τ* _{ a }(*m*, *k*) and *τ* _{ c }(*m*, *k*) combine to give *τ* _{1}(*m*, *k*) and *τ* _{ b }(*m*, *k*) and *τ* _{ d }(*m*, *k*) to *τ* _{2}(*m*, *k*). Only one island forms.

The death island of a pair of delay-coupled chaotic Rössler oscillators determined by the two critical curves *τ* _{1}(0, *k*) and *τ* _{2}(0, *K*) in Eq. (16). Now, *τ* _{ a }(*m*, *k*) and *τ* _{ c }(*m*, *k*) combine to give *τ* _{1}(*m*, *k*) and *τ* _{ b }(*m*, *k*) and *τ* _{ d }(*m*, *k*) to *τ* _{2}(*m*, *k*). Only one island forms.

Illustration of oscillation death by the time series of *x* _{1}(*t*) of two delay-coupled chaotic Rössler oscillators for two parameter sets: (a) (*τ* = 0.5, *K* = 1) (outside the oscillation death island) and (b) (*τ* = 1.0, *K* = 1) (inside the oscillation death island). They clearly show a dramatic difference between oscillatory state and oscillation death state after the transient. The same random initial conditions are chosen.

Illustration of oscillation death by the time series of *x* _{1}(*t*) of two delay-coupled chaotic Rössler oscillators for two parameter sets: (a) (*τ* = 0.5, *K* = 1) (outside the oscillation death island) and (b) (*τ* = 1.0, *K* = 1) (inside the oscillation death island). They clearly show a dramatic difference between oscillatory state and oscillation death state after the transient. The same random initial conditions are chosen.

Schematic illustrations for some regular networks: (a) a ring with nearest-neighbor coupling (*N* = 7), (b) an all-to-all network (*N* = 6), (c) a chain network, (d) a star network, (e) a grid network, and (f) a tree network. (c)-(f) are bipartite networks with nodes belonging to different parts shown by black and white colors, separately. A ring in (a) with even nodes is also bipartite.

Schematic illustrations for some regular networks: (a) a ring with nearest-neighbor coupling (*N* = 7), (b) an all-to-all network (*N* = 6), (c) a chain network, (d) a star network, (e) a grid network, and (f) a tree network. (c)-(f) are bipartite networks with nodes belonging to different parts shown by black and white colors, separately. A ring in (a) with even nodes is also bipartite.

The case for a ring with nearest-neighbor coupling. (a) *λ* _{ N } vs *N* (*N* is odd), showing *λ* _{ N } rapidly damps to *λ* _{ N } = –1. (b) The death islands determined by the four critical curves *τ* _{ a }(*m*, *k*), *τ* _{ b }(*m*, *K*), *τ* _{ c }(*m*, *k*), and *τ* _{ d }(*m*, *K*) in Eq. (11) for different *N*’s. Clearly, the death island decreases with the increase of (odd) *N* and approaches the smallest one as *N* → *∞*, and it remains unchanged and smallest for any even *N*.

The case for a ring with nearest-neighbor coupling. (a) *λ* _{ N } vs *N* (*N* is odd), showing *λ* _{ N } rapidly damps to *λ* _{ N } = –1. (b) The death islands determined by the four critical curves *τ* _{ a }(*m*, *k*), *τ* _{ b }(*m*, *K*), *τ* _{ c }(*m*, *k*), and *τ* _{ d }(*m*, *K*) in Eq. (11) for different *N*’s. Clearly, the death island decreases with the increase of (odd) *N* and approaches the smallest one as *N* → *∞*, and it remains unchanged and smallest for any even *N*.

Plot of the size ratio *R* in Eq. (21) as a function of *λ* _{ N } (–1 ≤ *λ* _{ N } < 0), which shows a rapid increase within the sensitive parameter region (on the left of the vertical dashed line) and a slow increase within the insensitive parameter region (on the right of the vertical dashed line). A crossover at *λ* _{ N } ≈ –0.9 is clear. As *λ* _{ N } > –0.9 for nearly all complex networks having quite different topologies, the size of death island keeps the largest and insensitively depends on the change of complex network structures.

Plot of the size ratio *R* in Eq. (21) as a function of *λ* _{ N } (–1 ≤ *λ* _{ N } < 0), which shows a rapid increase within the sensitive parameter region (on the left of the vertical dashed line) and a slow increase within the insensitive parameter region (on the right of the vertical dashed line). A crossover at *λ* _{ N } ≈ –0.9 is clear. As *λ* _{ N } > –0.9 for nearly all complex networks having quite different topologies, the size of death island keeps the largest and insensitively depends on the change of complex network structures.

(Color online) The case for all-to-all networks. (a) *λ* _{ N } vs *N*. (b) The death islands for *N* = 3 (black solid lines) and *N* = *∞* (red dashed lines), which indicate that all all-to-all delay-coupled oscillator networks have nearly the same death island. The numerical result represented by the open circles for *N* = 3 is in good agreement with the theoretic prediction.

(Color online) The case for all-to-all networks. (a) *λ* _{ N } vs *N*. (b) The death islands for *N* = 3 (black solid lines) and *N* = *∞* (red dashed lines), which indicate that all all-to-all delay-coupled oscillator networks have nearly the same death island. The numerical result represented by the open circles for *N* = 3 is in good agreement with the theoretic prediction.

The case for random networks of small sizes. Left column: (a)-(d) The sketch of random networks, which are generated by randomly adding edge(s) on a ring network (*N* = 16). The *λ* _{ N }’s of the generated networks are –1.0, –0.9782, –0.9627, and –0.9352, respectively. In (a), the network is bipartite illustrated with nodes by black and white colors. Right column: (e)-(h) The corresponding death islands of the left random networks. The numerical results represented by the open circles are all in good agreement with the theoretic prediction.

The case for random networks of small sizes. Left column: (a)-(d) The sketch of random networks, which are generated by randomly adding edge(s) on a ring network (*N* = 16). The *λ* _{ N }’s of the generated networks are –1.0, –0.9782, –0.9627, and –0.9352, respectively. In (a), the network is bipartite illustrated with nodes by black and white colors. Right column: (e)-(h) The corresponding death islands of the left random networks. The numerical results represented by the open circles are all in good agreement with the theoretic prediction.

The case for the well-known small-world (top) and scale-free networks (bottom). (a) The Watts-Strogatz networks. The parameter *p* refers to the rewiring probability. (b) The Newman-Watts networks. Here, *p* refers to the probability that new edges are added to the network. (c) The general SF networks. The preferential attachment probability is , with *k* _{ i } denoting the degree of node *i*, and *μ* representing the parameter that makes the scaling exponent *γ* tunable. , with *m* the number of edges that link new node each time (in the calculation, *m* = 3 is chosen.) (d) The general SF networks, with the preferential attachment probability . In all these plots, *λ* _{ N } in networks with the size *N* = 1000 is calculated and each data point is averaged over 100 realizations.

The case for the well-known small-world (top) and scale-free networks (bottom). (a) The Watts-Strogatz networks. The parameter *p* refers to the rewiring probability. (b) The Newman-Watts networks. Here, *p* refers to the probability that new edges are added to the network. (c) The general SF networks. The preferential attachment probability is , with *k* _{ i } denoting the degree of node *i*, and *μ* representing the parameter that makes the scaling exponent *γ* tunable. , with *m* the number of edges that link new node each time (in the calculation, *m* = 3 is chosen.) (d) The general SF networks, with the preferential attachment probability . In all these plots, *λ* _{ N } in networks with the size *N* = 1000 is calculated and each data point is averaged over 100 realizations.

The case for some other types of networks. (a) The Erdos-Renyi random networks, with *p* the random connect probability for each pair of nodes in the network. (b) The geographical networks, which are built on a 50 × 20 grid with the connect probability *P*(*i* → *j*) ∼ *exp*(−*βs* _{ ij }), and *s* _{ ij } = |*i* − *j*|. (c) The community networks, with the community number fixed at 4 and the edges inside communities increased with the connect probability *p*. (d) The community networks, with the edge probability inside community fixed at *p* = 0.5 and the number of communities *n* varying from 2 to 10. In all these calculations, the number of vertices *N* = 1000 is fixed and each data point is averaged over 100 different realizations.

The case for some other types of networks. (a) The Erdos-Renyi random networks, with *p* the random connect probability for each pair of nodes in the network. (b) The geographical networks, which are built on a 50 × 20 grid with the connect probability *P*(*i* → *j*) ∼ *exp*(−*βs* _{ ij }), and *s* _{ ij } = |*i* − *j*|. (c) The community networks, with the community number fixed at 4 and the edges inside communities increased with the connect probability *p*. (d) The community networks, with the edge probability inside community fixed at *p* = 0.5 and the number of communities *n* varying from 2 to 10. In all these calculations, the number of vertices *N* = 1000 is fixed and each data point is averaged over 100 different realizations.

(Color online) Demonstration of the insensitivity of delay-induced oscillation death on complex networks. The death islands of the WS networks with *p* = 0.0 (back thin line) and *p* = 1.0 (black thick line) and the general SF network with *γ* = 5.0 (dashed red line) exhibit nearly the same patterns with slight differences on the bottom. The numerical result represented by the open circles for the WS networks with *p* = 1.0 is in good agreement with the theoretic prediction.

(Color online) Demonstration of the insensitivity of delay-induced oscillation death on complex networks. The death islands of the WS networks with *p* = 0.0 (back thin line) and *p* = 1.0 (black thick line) and the general SF network with *γ* = 5.0 (dashed red line) exhibit nearly the same patterns with slight differences on the bottom. The numerical result represented by the open circles for the WS networks with *p* = 1.0 is in good agreement with the theoretic prediction.

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