zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Large-amplitude periodic solutions for differential equations with delayed monotone positive feedback. (English) Zbl 1247.34120

For scalar delay equations of the type

x ˙(t)=-μx(t)+f(x(t-1)),

the global attractor 𝒜 (within the class of solutions with not more than two zeros per time unit, in other words, within the corresponding level set of a zero-counting Liapunov functional) is studied. The situation is more complex than in previous work of Krisztin, Walther and Wu. Two spindle-like three dimensional subsets of 𝒜 (as described in the previous results) exist, joining equilibria ξ -2 <0<ξ 2 of the equation. There are also unstable equilibria ξ -1 ,ξ 1 satisfying ξ -2 <ξ -1 <0<ξ 1 <ξ 2 ; these correspond to the zero equilibrium in the quoted earlier work. It is shown that, for suitable μ>0 and C 1 -nonlinearity f, the global attractor 𝒜 contains, in addition, two periodic orbits with large-amplitude in the sense that their range includes the interval (ξ -1 ,ξ 1 ). Further, the dynamics on 𝒜 can be described completely. All the heteroclinic connections between different periodic orbits, or between periodic orbits and equilibria, which are not excluded by the discrete Liapunov functional do actually exist. The nonlinearities f are smoothed step functions, a feature that is probably important for the proofs (in that it allows explicit calculations), but not essential for the results. The given examples contribute much to the deeper understanding of the attractors for this type of infinite-dimensional dynamical systems.

MSC:
34K13Periodic solutions of functional differential equations
37C70Attractors and repellers, topological structure
37D05Hyperbolic orbits and sets
37L25Inertial manifolds and other invariant attracting sets
37L45Hyperbolicity; Lyapunov functions (infinite-dimensional dissipative systems)
34K25Asymptotic theory of functional-differential equations
34D45Attractors
34C45Invariant manifolds (ODE)
34C37Homoclinic and heteroclinic solutions of ODE
34K21Stationary solutions of functional-differential equations
References:
[1]Chen Y., Wu J.: Minimal instability and unstable set of a phase-locked periodic orbit in a delayed neural network. Physica D 134, 185–199 (1999) · Zbl 0942.34062 · doi:10.1016/S0167-2789(99)00111-6
[2]Chen Y., Wu J.: Existence and attraction of a phase-locked oscillation in a delayed network of two neurons. Differ. Integral Equ. 14, 1181–1236 (2001)
[3]Chen Y., Krisztin T., Wu J.: Connecting orbits from synchronous periodic solutions in phase-locked periodic solutions in a delay differential system. J. Diff. Equ. 163, 130–173 (2000) · Zbl 0955.34058 · doi:10.1006/jdeq.1999.3724
[4]Chen Y., Yi T., Wu J.: Periodic solutions and the global attractor in a system of delay differential equations. SIAM J. Math. Anal. 42, 24–63 (2010) · Zbl 1229.34110 · doi:10.1137/080725283
[5]Computer Assisted Proofs in Dynamics group, a C++ package for rigorous numerics, http://capd.wsb-nlu.edu.pl
[6]Diekmann O., van Gils S.A., Verduyn Lunel S.M., Walther H.-O.: Delay Equations. Functional, Complex, and Nonlinear Analysis. Springer, New York (1995)
[7]Erneux T.: Applied Delay Differential Equations. Springer, New York (2009)
[8]Gerstner W., Kistler W.M.: Spiking Neuronmodels, Single Neurons, Populations, Plasticity. Cambridge University Press, Cambridge (2002)
[9]Hale J.K.: Ordinary Differential Equations. Interscience, New York (1969)
[10]Hale J.K.: Asymptotic Behavior of Dissipative Systems. American Mathematical Society, Providence (1988)
[11]Hale J.K., Verduyn Lunel S.M.: Introduction to Functional-Differential Equations. Springer, New York (1993)
[12]Kennedy, B.: Stability and instability for periodic solutions of delay equations with steplike feedback. In: preparation
[13]Krisztin, T.: The unstable set of zero and the global attractor for delayed monotone positive feedback. Discr. Contin. Dyn. Syst., Added Volume, 229–240 (2000)
[14]Krisztin T.: Unstable sets of periodic orbits and the global attractor for delayed feedback. In: Faria, T., Freitas, P. (eds) Topics in Functional Differential and Difference Equations, pp. 267–296. American Mathematical Society, Providence (2001)
[15]Krisztin T.: Global dynamics of delay differential equations. Period. Math. Hung. 56, 83–95 (2008) · Zbl 1164.34037 · doi:10.1007/s10998-008-5083-x
[16]Krisztin T., Walther H.-O.: Unique periodic orbits for delayed positive feedback and the global attractor. J. Dyn. Differ. Equ. 13, 1–57 (2001) · Zbl 1008.34061 · doi:10.1023/A:1009091930589
[17]Krisztin, T., Wu, J.: The global structure of an attracting set. In: preparation
[18]Krisztin T., Walter H.-O., Wu J.: Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback. American Mathematical Society, Providence (1999)
[19]Lani-Wayda B.: Persistence of Poincaré mappings in functional differential equations (with application to structural stability of complicated behavior). J. Dyn. Differ. Equ. 7, 1–71 (1995) · Zbl 0832.34082 · doi:10.1007/BF02218814
[20]Mallet-Paret J.: Morse decompositions for delay-differential equations. J. Differ. Equ. 72, 270–315 (1988) · Zbl 0648.34082 · doi:10.1016/0022-0396(88)90157-X
[21]Mallet-Paret J., Sell G.R.: Systems of differential delay equations: Floquet multipliers and discrete Lyapunov Functions. J. Differ. Equ. 125, 385–440 (1996a) · Zbl 0849.34055 · doi:10.1006/jdeq.1996.0036
[22]Mallet-Paret J., Sell G.R.: The Poincaré–Bendixson theorem for monotone cyclic feedback systems with delay. J. Differ. Equ. 125, 441–489 (1996b) · Zbl 0849.34056 · doi:10.1006/jdeq.1996.0037
[23]McCord C., Mischaikow K.: On the global dynamics of attractors for scalar delay equations. J. Am. Math. Soc. 9, 1095–1133 (1996) · Zbl 0861.58023 · doi:10.1090/S0894-0347-96-00207-X
[24]Polner M.: Morse decomposition for delay differential equations with positive feedback. Nonlinear Anal. 48, 377–397 (2002) · Zbl 1003.34065 · doi:10.1016/S0362-546X(00)00191-7
[25]Sard A.: The measure of the critical values of differentiable maps. Bull. Am. Math. Soc. 48, 883–890 (1942) · Zbl 0063.06720 · doi:10.1090/S0002-9904-1942-07811-6
[26]Smith H.L.: Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. American Mathematical Society, Providence (1995)
[27]Stoffer D.: Delay equations with rapidly oscillating stable periodic solutions. J. Dyn. Differ. Equ. 20, 201–238 (2008) · Zbl 1204.34090 · doi:10.1007/s10884-006-9068-4
[28]Walther H.-O.: The 2-dimensional attractor of x’(t) = x(t) + f(x(t 1)). Mem. Am. Math. Soc. 113(544), 1–76 (1995)
[29]Walther H.-O.: Contracting return maps for some delay differential equations. In: Faria, T., Freitas, P. (eds) Topics in Functional Differential and Difference Equations, pp. 349–360. American Mathematical Society, Providence (2001)
[30]Walther H.-O.: The singularities of an attractor of a delay differential equation. Funct. Differ. Equ. 5, 513–548 (1998)
[31]Wu J.: Introduction to Neural Dynamics and Signal Transmission Delay. Walter de Gruyter &amp; Co., Berlin (2001)