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Trajectories joining critical points
1982
Journal of Differential Equations
The o-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 ...
doi:10.1016/0022-0396(82)90024-9
fatcat:zteilm6o4nbdjcldwwqqgf7bl4
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 ...
Asymptotic behavior of coupled dynamical systems with multiscale aspects
2010
Journal of Differential Equations
We show several results ranging from 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). ...
doi:10.1016/j.jde.2009.06.014
fatcat:hk5mv6cqgfdk3gyftorn44mhe4
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 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). ...
arXiv:0904.0397v1
fatcat:6jyazv64wjec3ox5n3fifgd6lm
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 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 ...
doi:10.1016/s0020-7225(98)00092-5
fatcat:h5grb6cfjzfzpn2li33u37pfza
Electrical circuits with chaotic behavior
1987
Proceedings of the IEEE
It can be proved by using the stored energy as a 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] . ...
doi:10.1109/proc.1987.13846
fatcat:m3igbonxl5eorgnleqfocnbt3m
Error Bounds and Applications for Stochastic Approximation with Non-Decaying Gain
[article]
2020
arXiv
pre-print
The 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 → ∞. ...
arXiv:2003.07357v1
fatcat:ebwhbynunzbvdccjwn73txeooi
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 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' ...
doi:10.1002/0471231509.ch2
fatcat:kgi2e5kjbjd3lhhweilppar4wm
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 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. ...
doi:10.1142/s0218127408021002
fatcat:7rqniqmfqbas7frg5yg6zszawq
Renormalization Group as a Probe for Dynamical Systems
2011
Journal of Physics, Conference Series
in the 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. ...
doi:10.1088/1742-6596/319/1/012017
fatcat:4bx3c6e4tbcstl46f2l7ohh6by
The control of chaos: theory and applications
2000
Physics reports
In particular, special care should be exercised when dealing 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 ...
doi:10.1016/s0370-1573(99)00096-4
fatcat:ukfcbglvgbguri7efiiz6jxdqu
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 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. ...
doi:10.5194/npg-21-61-2014
fatcat:yrj3okzq7fa2vgfk6vj2xjeumu
Planar dynamics and control of tethered satellite systems
1994
The governing nonlinear, 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. ...
doi:10.14288/1.0088399
fatcat:cktqvdourbgr3hulbgxux7dsce
Coarse-graining for gradient systems and Markov processes
2021
The first three parts of the thesis deal 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 ...
doi:10.18452/23529
fatcat:kgdugr7545cwpdh3lu4eewdffa
Hopf bifurcations in magnetoconvection in the presence of sidewalls
1993
Our model partial differential equations together 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 ...
doi:10.14288/1.0079821
fatcat:z7j73ivrdzdv7k5xgjt2k6fbge