• Previous Article
    Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I —
  • CPAA Home
  • This Issue
  • Next Article
    Nonhomogeneous polyharmonic elliptic problems at critical growth with symmetric data
June  2003, 2(2): 187-209. doi: 10.3934/cpaa.2003.2.187

Attractors in continuous –time switching networks

1. 

Department of Mathematical Sciences, Montana State University, Bozeman, MT 59717, United States

Received  February 2002 Revised  January 2003 Published  March 2003

We consider a system of equations with discontinuous right hand side, which arise as models of gene and neural networks. We study attractors in $R^4$ which lie in a set of orthants in the form of figure eight. We find that if the attractor is symmetric with respect to these two loops, then the only possible attractor is a periodic orbit which traverses both loops once. We show that without the symmetry the set of admissible attractors include periodic orbits which follow one loop $k$ times and other loop once, for any $k$. However, we also show that no trajectory in an attractor can traverse both loops more then once in a row.
Citation: Tomáš Gedeon. Attractors in continuous –time switching networks. Communications on Pure & Applied Analysis, 2003, 2 (2) : 187-209. doi: 10.3934/cpaa.2003.2.187
[1]

Zhigang Zeng, Tingwen Huang. New passivity analysis of continuous-time recurrent neural networks with multiple discrete delays. Journal of Industrial & Management Optimization, 2011, 7 (2) : 283-289. doi: 10.3934/jimo.2011.7.283

[2]

Shui-Nee Chow, Xiaojing Ye, Hongyuan Zha, Haomin Zhou. Influence prediction for continuous-time information propagation on networks. Networks & Heterogeneous Media, 2018, 13 (4) : 567-583. doi: 10.3934/nhm.2018026

[3]

Fritz Colonius, Guilherme Mazanti. Decay rates for stabilization of linear continuous-time systems with random switching. Mathematical Control & Related Fields, 2019, 9 (1) : 39-58. doi: 10.3934/mcrf.2019002

[4]

Ping Chen, Haixiang Yao. Continuous-time mean-variance portfolio selection with no-shorting constraints and regime-switching. Journal of Industrial & Management Optimization, 2020, 16 (2) : 531-551. doi: 10.3934/jimo.2018166

[5]

Simone Göttlich, Stephan Martin, Thorsten Sickenberger. Time-continuous production networks with random breakdowns. Networks & Heterogeneous Media, 2011, 6 (4) : 695-714. doi: 10.3934/nhm.2011.6.695

[6]

Ying Sue Huang. Resynchronization of delayed neural networks. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 397-401. doi: 10.3934/dcds.2001.7.397

[7]

Tingting Su, Xinsong Yang. Finite-time synchronization of competitive neural networks with mixed delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3655-3667. doi: 10.3934/dcdsb.2016115

[8]

Cheng-Hsiung Hsu, Suh-Yuh Yang. Traveling wave solutions in cellular neural networks with multiple time delays. Conference Publications, 2005, 2005 (Special) : 410-419. doi: 10.3934/proc.2005.2005.410

[9]

Tatyana S. Turova. Structural phase transitions in neural networks. Mathematical Biosciences & Engineering, 2014, 11 (1) : 139-148. doi: 10.3934/mbe.2014.11.139

[10]

Joon Kwon, Panayotis Mertikopoulos. A continuous-time approach to online optimization. Journal of Dynamics & Games, 2017, 4 (2) : 125-148. doi: 10.3934/jdg.2017008

[11]

Hanqing Jin, Xun Yu Zhou. Continuous-time portfolio selection under ambiguity. Mathematical Control & Related Fields, 2015, 5 (3) : 475-488. doi: 10.3934/mcrf.2015.5.475

[12]

Wenlian Lu, Fatihcan M. Atay, Jürgen Jost. Consensus and synchronization in discrete-time networks of multi-agents with stochastically switching topologies and time delays. Networks & Heterogeneous Media, 2011, 6 (2) : 329-349. doi: 10.3934/nhm.2011.6.329

[13]

Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020200

[14]

Ruoxia Li, Huaiqin Wu, Xiaowei Zhang, Rong Yao. Adaptive projective synchronization of memristive neural networks with time-varying delays and stochastic perturbation. Mathematical Control & Related Fields, 2015, 5 (4) : 827-844. doi: 10.3934/mcrf.2015.5.827

[15]

Jianping Zhou, Yamin Liu, Ju H. Park, Qingkai Kong, Zhen Wang. Fault-tolerant anti-synchronization control for chaotic switched neural networks with time delay and reaction diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020357

[16]

Udhayakumar Kandasamy, Rakkiyappan Rajan. Hopf bifurcation of a fractional-order octonion-valued neural networks with time delays. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2537-2559. doi: 10.3934/dcdss.2020137

[17]

Fabian Rüffler, Volker Mehrmann, Falk M. Hante. Optimal model switching for gas flow in pipe networks. Networks & Heterogeneous Media, 2018, 13 (4) : 641-661. doi: 10.3934/nhm.2018029

[18]

Benedict Leimkuhler, Charles Matthews, Tiffany Vlaar. Partitioned integrators for thermodynamic parameterization of neural networks. Foundations of Data Science, 2019, 1 (4) : 457-489. doi: 10.3934/fods.2019019

[19]

Ricai Luo, Honglei Xu, Wu-Sheng Wang, Jie Sun, Wei Xu. A weak condition for global stability of delayed neural networks. Journal of Industrial & Management Optimization, 2016, 12 (2) : 505-514. doi: 10.3934/jimo.2016.12.505

[20]

Benedetta Lisena. Average criteria for periodic neural networks with delay. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 761-773. doi: 10.3934/dcdsb.2014.19.761

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (29)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]