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### Trajectories joining critical points

Zvi Artstein, Marshall Slemrod
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  ...

### 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

Hedy Attouch, Marc-Olivier Czarnecki
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).  ...

### Asymptotic behavior of coupled dynamical systems with multiscale aspects [article]

Hedy Attouch, Marc-Olivier Czarnecki
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).  ...

### Regular and chaotic dynamics of a rotational machine with a centrifugal governor

Z.-M. Ge, H.-S. Yang, H.-H. Chen, H.-K. Chen
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  ...

### Electrical circuits with chaotic behavior

M.J. Hasler
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] .  ...

### Error Bounds and Applications for Stochastic Approximation with Non-Decaying Gain [article]

Jingyi Zhu
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 → ∞.  ...

### Chemical Reaction Dynamics: Many-Body Chaos and Regularity [chapter]

Tamiki Komatsuzaki, R. Stephen Berry
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'  ...

### RANK ONE CHAOS: THEORY AND APPLICATIONS

QIUDONG WANG, ALI OKSASOGLU
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.  ...

### Renormalization Group as a Probe for Dynamical Systems

Amartya Sarkar, J K Bhattacharjee
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.  ...

### The control of chaos: theory and applications

S Boccaletti
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  ...

### Stochastic electron motion driven by space plasma waves

G. V. Khazanov, A. A. Tel'nikhin, T. K. Kronberg
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.  ...

### 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.  ...

### Coarse-graining for gradient systems and Markov processes

Artur Stephan, Humboldt-Universität Zu Berlin
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  ...

### Hopf bifurcations in magnetoconvection in the presence of sidewalls

Hamid Zangeneh
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  ...