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• 1
Electronic Resource
Springer
Archive for rational mechanics and analysis 62 (1976), S. 209-235
ISSN: 1432-0673
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Type of Medium: Electronic Resource
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• 2
Electronic Resource
Springer
Archive for rational mechanics and analysis 129 (1995), S. 245-304
ISSN: 1432-0673
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract For linear scalar parabolic equations such as $$u_t = u_{xx} + a(t,x)u_x + b(t,x)u$$ on a finite interval 0≦x≦π, with various boundary conditions, we obtain canonical Floquet solutions u n (t, x). These solutions are characterized by the property that z(u n (t, x))=n for all tεℝ, where z(·) denotes the zero crossing (lap) number of Matano. The coefficients a(t, x) and b(t, x) are not assumed to be periodic in t, but if they are, the solutions u n (t, x) reduce to the standard Floquet solutions. Our results may naturally be expressed in the language of linear skew product flows. In this context, we obtain for each N≧1 an exponential dichotomy between the bundles span {u n (·,·)} n =1/N and $$\overline {span} \{ u_n ( \cdot , \cdot )\} _{n = N + 1}^\infty$$ .
Type of Medium: Electronic Resource
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• 3
Electronic Resource
Springer
Archive for rational mechanics and analysis 59 (1975), S. 159-188
ISSN: 1432-0673
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Type of Medium: Electronic Resource
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• 4
Electronic Resource
Springer
Journal of dynamics and differential equations 6 (1994), S. 543-582
ISSN: 1572-9222
Keywords: Center-like manifolds ; invariant foliations
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract We study the existence and smoothness of global center, center-stable, and center-unstable manifolds for skew-product flows. Smooth invariant foliations to the center stable and center unstable manifolds are also discussed.
Type of Medium: Electronic Resource
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• 5
Electronic Resource
Springer
Journal of dynamics and differential equations 7 (1995), S. 375-407
ISSN: 1572-9222
Keywords: Lorenz type attractors ; conventional Floquet exponents ; hyperbolicity conditions ; saddle-node bifurcations
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract Lorenz type attractors are found from a codimension one bifurcation of a system on the boundary of Morse-Smale systems. Conditions of their emerging are formulated in terms of conventional Floquet exponents of homoclinic orbits—a new characteristic of homoclinic orbits at the bifurcation moment.
Type of Medium: Electronic Resource
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• 6
Electronic Resource
Springer
Journal of dynamics and differential equations 2 (1990), S. 177-244
ISSN: 1572-9222
Keywords: homoclinic orbit ; period doubling ; pathfollowing ; global bifurcation ; resonance ; 34C15 ; 34C35 ; 58F14
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract We consider a bifurcation of homoclinic orbits, which is an analogue of period doubling in the limit of infinite period. This bifurcation can occur in generic two parameter vector fields when a homoclinic orbit is attached to a stationary point with resonant eigenvalues. The resonance condition requires the eigenvalues with positive/negative real part closest to zero to be real, simple, and equidistant to zero. Under an additional global twist condition, an exponentially flat bifurcation of double homoclinic orbits from the primary homoclinic branch is established rigorously. Moreover, associated period doublings of periodic orbits with almost infinite period are detected. If the global twist condition is violated, a resonant side switching occurs. This corresponds to an exponentially flat bifurcation of periodic saddle-node orbits from the homoclinic branch.
Type of Medium: Electronic Resource
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• 7
Electronic Resource
Springer
Journal of dynamics and differential equations 3 (1991), S. 361-379
ISSN: 1572-9222
Keywords: Dynamical systems ; numerically computed orbits ; shadowing ; pseudo-orbits ; round-off ; expansiveness
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract A finite time version of the shadowing theorem is used to develop a procedure to determine the accuracy of numerically computed orbits of one-dimensional maps. The procedure works forward. After any given number of iterates, we can decide whether our theorem applies and, if it does, we can estimate how far the computed orbit is from a true orbit.
Type of Medium: Electronic Resource
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• 8
Electronic Resource
Springer
Journal of dynamics and differential equations 1 (1989), S. 3-43
ISSN: 1572-9222
Keywords: Differential delay equations ; singular perturbations ; transition layers
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract Transition layers arising from square-wave-like periodic solutions of a singularly perturbed delay differential equation are studied. Such transition layers correspond to heteroclinic orbits connecting a pair of equilibria of an associated system of transition layer equations. Assuming a monotonicity condition in the nonlinearity, we prove these transition layer equations possess a unique heteroclinic orbit, and that this orbit is monotone. The proof involves a global continuation for heteroclinic orbits.
Type of Medium: Electronic Resource
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• 9
Electronic Resource
Springer
Annali di matematica pura ed applicata 94 (1972), S. 377-386
ISSN: 1618-1891
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Summary Necessary and sufficient conditions for integral boundedness are given in terms of a Liapunov function are given the construction of the Liapunov function is a modification of the Okamura function. Similar results are also given for the extendability of solutions under integral perturbations.
Type of Medium: Electronic Resource
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• 10
Book
Berlin u.a. :Springer,
Title: Methods of bifurcation theory; 251
Author: Chow, Shui-Nee
Publisher: Berlin u.a. :Springer,
Year of publication: 1982
Pages: 515 S.
Series Statement: ¬Die¬ Grundlehren der mathematischen Wissenschaften 251
Type of Medium: Book
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