Bifurcations in Glass networks

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 15, Issue 2, 2005, Pages 395-423

  Killough, D B ^a   Edwards, R ^a  

^a UNIVERSITY OF VICTORIA   (Canada)

Author keywords

Border collision; C bifurcation; Discontinuous vector field; Glass network; Piecewise linear; Routes to chaos

Indexed keywords

APPROXIMATION THEORY; BIOCHEMISTRY; CHAOS THEORY; GENES; MATRIX ALGEBRA; NEURAL NETWORKS;

BORDER-COLLISION; C-BIFURCATION; DISCONTINOUS VECTOR FIELDS; GLASS NETWORKS; PIECEWISE-LINEAR; ROUTES TO CHAOS;

BIFURCATION (MATHEMATICS);

SIGMOIDEA;

EID: 18644382642     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127405012302     Document Type: Article

References (24)

1. di Bernardo, M., Budd, C. & Champneys, A. R. [1998] "Grazing, skipping and sliding: Analysis of the non-smooth dynamics of the DC/DC buck converter," Nonlinearity 11, 859-890.
2. di Bernardo, M., Feigin, M. I., Hogan, S. J. & Homer, M. E. [1999] "Local analysis of C-bifurcations in n-dimensional piecewise-smooth dynamical systems," Chaos Solit. Fract. 10, 1881-1908.
3. di Bernardo, M., Budd, C. J. & Champneys, A. R. [2001a] "Corner collision implies border-collision bifurcation," Physica D154, 171-194.
4. di Bernardo, M., Budd, C. J. & Champneys, A. R. [2001b] "Grazing and border-collision in piecewise-smooth systems: A unified analytical framework," Phys. Rev. Lett. 86, 2553-2556.
5. di Bernardo, M., Kowalczyk, P. & Nordmark, A. [2002] "Bifurcations of dynamical systems with sliding: Derivation of normal-form mappings." Physica D170, 175-205.
6. Edwards, R., Beuter, A. & Glass, L. [1999] "Parkinsonian tremor and simplification in network dynamics," Bull. Math. Biol. 51, 157-177.
7. Edwards, R. [2000] "Analysis of continuous-time switching networks," Physica D146, 165-199.
8. Edwards, R. [2001] "Chaos in neural and gene networks with hard switching," Diff. Eqs. Dyn. Syst. 9, 187-220.
9. Edwards, R., Siegelmann, H. T., Aziza, K. & Glass, L. [2001] "Symbolic dynamics and computation in model gene networks," Chaos 11, 160-169.
10. Feigin, M. I. [1995] "The increasingly complex structure of the bifurcation tree of a piecewise-smooth system," J. Appl. Math. Mech. 59, 853-863.
11. Filippov, A. F. [1988] Differential Equations with Discontinuous Righthand Sides (Kulwer Academic Publishers, Dordrecht).
12. Glass, L. [1975] "Combinatorial and topological methods in nonlinear chemical kinetics," J. Chem. Phys. 63, 1325-1335.
13. Glass, L. & Pasternack, J. S. [1978] "Prediction of limit cycles in mathematical models of biological oscillations," Bull. Math. Biol. 40, 27-44.
14. Guckenheimer, J. & Holmes, P. [1983] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, NY).
15. Kappler, K., Edwards, R. & Glass, L. [2003] "Dynamics in high-dimensional model gene networks," Sign. Process. 83, 789-798.
16. Lewis, J. E. & Glass, L. [1991] "Steady states, limit cycles, and chaos in models of complex biological networks," Int. J. Bifurcation and Chaos 1, 477-483.
17. Lewis, J. E. & Glass, L. [1992] "Nonlinear dynamics and symbolic dynamics of neural networks." Neural Comput. 4, 621-642.
18. Maggio, G. M., di Bernardo, M. & Kennedy, M. P. [2000] "Nonsmooth bifurcations in a piecewise-linear model of the Colpitts oscillator," IEEE Trans. Circuits Syst.-I: Fund. Th. Appl. 47, 1160-1177.
19. Manneville, P. [1990] Dissipative Structures and Weak Turbulence (Academic Press, San Diego).
20. Mestl, T., Plahte, E. & Omholt, S. W. [1995] "Periodic solutions in systems of piecewise-linear differential equations," Dyn. Stab. Syst. 10, 179-193.
21. Mestl, T., Lemay, C. & Glass, L. [1996] "Chaos in high-dimensional neural and gene networks," Physica D98, 33-52.
22. Nusse, H. E., Ott, E. & Yorke, J. A. [1994] "Border-collision bifurcations: An explanation for observed bifurcation phenomena," Phys. Rev. E49, 1073-1076.
23. Strogatz, S. H. [1994] Nonlinear Dynamics and Chaos: With Applications to Physics, Biology. Chemistry, and Engineering (Perseus Books, Cambridge, MA).
24. Yuan, G., Banerjee, S., Ott, E. & Yorke, J. A. [1998] "Border-collision bifurcations in the buck converter," IEEE Trans. Circuits Syst.-I: Fund. Th. Appl. 45, 707-716.