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Trajectories joining critical points

1982
*
Journal of Differential Equations
*

The o-

doi:10.1016/0022-0396(82)90024-9
fatcat:zteilm6o4nbdjcldwwqqgf7bl4
*limit**set*of CJ, denoted w(U) is the*set*of all*limits*z = lim U(t,) for sequences t, -+ co. ... If U is a full orbit then its o-*limit**set*is the*set*a(U) of all*limits*z = lim U(tJ for sequences t, -+ --co. ... This would contradict the existence of the*Liapunov**function*V since a Li'apunov*function*is always constant on an a-*limit**set*, and an a-*limit**set*is positively invariant. weakiy joining orbit between ...##
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Page 2038 of Mathematical Reviews Vol. 52, Issue 6
[page]

1976
*
Mathematical Reviews
*

K. 14513

*Locating**limit**sets**with**weak**nonautonomous**Liapunov**functions*. Math. Systems Theory 8 (1974/75), no. 3, 228-234. ... For the system x’=f(x, t), where f is continuous on a cylinder D=P x [8, 0) and P is an open, connected subset of R", the author extends two theorems due to LaSalle and the reviewer on*locating**limit**sets*...##
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Asymptotic behavior of coupled dynamical systems with multiscale aspects

2010
*
Journal of Differential Equations
*

We show several results ranging from

doi:10.1016/j.jde.2009.06.014
fatcat:hk5mv6cqgfdk3gyftorn44mhe4
*weak*ergodic to strong convergence of the trajectories. ... We study the asymptotic behavior, as time variable t goes to +∞, of*nonautonomous*dynamical systems involving multiscale features. ... For each t > 0*set*X(t) = 1 t t 0 x(s) ds. Every*weak*-*limit*point of X(.) belongs to S = (A + N C ) −1 (0). Proof. Let t n → +∞ and suppose X(t n ) X ∞ (*weak*convergence in H). ...##
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Asymptotic behavior of coupled dynamical systems with multiscale aspects
[article]

2009
*
arXiv
*
pre-print

We study the asymptotic behavior, as time t goes to infinity, of

arXiv:0904.0397v1
fatcat:6jyazv64wjec3ox5n3fifgd6lm
*nonautonomous*dynamical systems involving multiscale features. ... For each t > 0*set*X(t) = 1 t t 0 x(s)ds. Every*weak**limit*point of X(.) belongs to S. Proof of Lemma 2.2. Let t n → +∞ and suppose X(t n ) ⇀ X ∞ (*weak*convergence in H). ... By using energetic*Liapunov*methods, under the additional growth condition on β, namelyβ ≤ kβ, we prove an asymptotic*weak*convergence result (theorem 3.1). ...##
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Regular and chaotic dynamics of a rotational machine with a centrifugal governor

1999
*
International Journal of Engineering Science
*

The incremental harmonic balance (IHB) method combined

doi:10.1016/s0020-7225(98)00092-5
fatcat:h5grb6cfjzfzpn2li33u37pfza
*with*the multi-variable Floquet theory has been eectively applied to obtain the steady state responses of the three-dimensional*nonautonomous*system ... The dynamic behavior of a rotational machine*with*centrifugal governor which is subjected to two dierent forms of external disturbance is studied in this paper. ... Construct the quadratic Lyapunov*function*candidate in the form Vx,y,z A 11 x 2 A 22 y 2 A 33 z 2 2A 12 xy 2A 13 xz À 2yzX The derivative of V*with*respect to t along the trajectories of the system is ...##
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Electrical circuits with chaotic behavior

1987
*
Proceedings of the IEEE
*

It can be proved by using the stored energy as a

doi:10.1109/proc.1987.13846
fatcat:m3igbonxl5eorgnleqfocnbt3m
*Liapunov**function*[3] . Therefore, it does not make sense to talk about synchronization in this case. ... OTHER*NONAUTONOMOUS*CIRCUITS There are other*nonautonomous*circuits*with*chaotic behavior, in particular the circuits that are described by the Duffing equation [24] . ...##
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Error Bounds and Applications for Stochastic Approximation with Non-Decaying Gain
[article]

2020
*
arXiv
*
pre-print

The

arXiv:2003.07357v1
fatcat:ebwhbynunzbvdccjwn73txeooi
*setting*is to minimize a sequence of scalar-valued loss*functions*f_k(·) at sampling times τ_k or to*locate*the root of a sequence of vector-valued*functions*g_k(·) at τ_k*with*respect to a parameter ... The*weak*convergence*limit*of the continuous interpolation of θ̂_k is shown to follow the trajectory of a non-autonomous ordinary differential equation. ... The*weak*convergence*limit*≤ m (t)} as n → ∞. ...##
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Chemical Reaction Dynamics: Many-Body Chaos and Regularity
[chapter]

2003
*
Advances in Chemical Physics
*

Described in terms of a multidimensional surface of internal energy as a

doi:10.1002/0471231509.ch2
fatcat:kgi2e5kjbjd3lhhweilppar4wm
*function*of the*locations*of the atomic nuclei, this model has the reacting system go from one local minimum across a saddle in ... They revealed, by analyses of local*Liapunov**functions*and Kolmogorov entropies, that when systems have just enough energy to pass through the transition state, the systems' trajectories become collimated ... Finally, the Lie transforms on*functions*f generated by a*nonautonomous*''Hamiltonian'' W can be represented as Note again that one might follow this by putting ''p'' and ''q'' into the ''Hamiltonian' ...##
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RANK ONE CHAOS: THEORY AND APPLICATIONS

2008
*
International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
*

Observe that the attractive
strength of Hopf

doi:10.1142/s0218127408021002
fatcat:7rqniqmfqbas7frg5yg6zszawq
*limit*cycles can be made as*weak*as desired. ... This could be achieved*with*either a sufficiently small λ (*weak*stability) and/or a sufficiently large B. ...##
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Renormalization Group as a Probe for Dynamical Systems

2011
*
Journal of Physics, Conference Series
*

in the

doi:10.1088/1742-6596/319/1/012017
fatcat:4bx3c6e4tbcstl46f2l7ohh6by
*weak*nonlinearity*limit*. ...*Liapunov**function*for this system is defined as a continuously differentiable, real-valued*function*V (x) such that (a) V (x) > 0 ∀ x = x * i.e. ... Actually it so happens that at γ = 1, another real fixed point, S, given by ((1 + 1/γ) 1/2 , (1 − 1/γ) 1/2 ) emerges and it coincides*with*R. ...##
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The control of chaos: theory and applications

2000
*
Physics reports
*

In particular, special care should be exercised when dealing

doi:10.1016/s0370-1573(99)00096-4
fatcat:ukfcbglvgbguri7efiiz6jxdqu
*with*pole placement technique for*nonautonomous*systems. ... This is done by*locating*pairs of points*with*same values of R in the computer representation of the coding*function*R(x) and choosing the one that yields the smallest value of x. ... Alice consists of two identical chaotic systems where is a*set*of control parameters chosen in such a way as to produce chaos, x 1 , x 2 are two D-dimensional vectors (D53) and f is a nonlinear*function*...##
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Stochastic electron motion driven by space plasma waves

2014
*
Nonlinear Processes in Geophysics
*

</strong> Stochastic motion of relativistic electrons under conditions of the nonlinear resonance interaction of particles

doi:10.5194/npg-21-61-2014
fatcat:yrj3okzq7fa2vgfk6vj2xjeumu
*with*space plasma waves is studied. ... Z, Z is the*set*of all integers. ... The*function*w(u, t) is a differentiable*function*supported in {U }*with*the norming u∈{U } w(u, t)du = 1, (80) where {U } is a range of the variable u. ...##
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Planar dynamics and control of tethered satellite systems

1994

The governing nonlinear,

doi:10.14288/1.0088399
fatcat:cktqvdourbgr3hulbgxux7dsce
*nonautonomous*and coupled equations of motion are obtained using the Lagrange procedure. ... The acceptable steady state error*limits*are*set*at ± 1 cm for offset positions and ± 0.5°for the tether angles. ... This*sets*the stage for an effective controller design. As pointed out before, the governing equations of motion are coupled, non linear and*nonautonomous*. ...##
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Coarse-graining for gradient systems and Markov processes

2021

The first three parts of the thesis deal

doi:10.18452/23529
fatcat:kgdugr7545cwpdh3lu4eewdffa
*with*fast-reaction*limits*for reaction systems and reaction-diffusion systems. ... This thesis deals*with*coarse-graining for gradient systems and Markov processes. ... Forschungsgemeinschaft (DFG) through the Collaborative Research Center SFB 1114 "Scaling Cascades in Complex Systems" (Project no. 235221301), Subproject C05 "Effective models for materials and interfaces*with*...##
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Hopf bifurcations in magnetoconvection in the presence of sidewalls

1993

Our model partial differential equations together

doi:10.14288/1.0079821
fatcat:z7j73ivrdzdv7k5xgjt2k6fbge
*with*the boundary conditions have two reflection symmetries. ... We assume that the fluid flow is two-dimensional, and consider the effects of sidewalls*with*stress-free boundary conditions. ... where 0 is not an even*function*of x, so that there is no longer a reflection symmetry under X -÷ -X. Bibliography ...