Routes to chaos in a delay-differential equation system modelling a passive optical resonator

1989 ◽  
Vol 139 (3-4) ◽  
pp. 133-140 ◽  
Author(s):  
F. Kaiser ◽  
D. Merkle
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Equation System ◽  
Optical Resonator ◽  
System Modelling ◽  
Differential Equation System ◽  
Routes To Chaos ◽  
Delay Differential

NONLINEAR OSCILLATIONS IN BUSINESS CYCLE MODEL WITH TIME LAGS

2000 ◽  
Vol 03 (03) ◽  
pp. 603-604 ◽  
Author(s):  
ADAM KRAWIEC ◽  
MAREK SZYDŁOWSKI ◽  
JANUSZ TOBOŁA
Keyword(s):  
Differential Equation ◽  
Business Cycle ◽  
Nonlinear Oscillations ◽  
Equation System ◽  
Hopf Bifurcations ◽  
Time Lags ◽  
Differential Equation System ◽  
Cycle Model ◽  
Business Cycle Model ◽  
Delay Differential

We analyse the Hopf bifurcations in the Kaldor–Kalecki business cycle model represented by a delay differential equation system.

scholarly journals Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 105
Author(s):  
Lokesh Singh ◽  
Dhirendra Bahuguna
Keyword(s):  
Differential Equation ◽  
Growth Rate ◽  
Invariant Manifold ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Stable Manifold ◽  
Exponential Dichotomy ◽  
Delay Differential ◽  
Stable Invariant ◽  
Nonuniform Exponential Dichotomy

In this article, we construct a C1 stable invariant manifold for the delay differential equation x′=Ax(t)+Lxt+f(t,xt) assuming the ρ-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the C1 perturbation, f(t,xt), and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation f(t,xt).

scholarly journals Transient Dynamics in the Random Growth and Reset Model

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 306
Author(s):  
Tamás S. Biró ◽  
Lehel Csillag ◽  
Zoltán Néda
Keyword(s):  
Differential Equation ◽  
Mean Field ◽  
Equation System ◽  
Transient Dynamics ◽  
Type Model ◽  
Discrete Version ◽  
Differential Equation System ◽  
Practical Applications ◽  
Random Growth ◽  
Recursive Integration

A mean-field type model with random growth and reset terms is considered. The stationary distributions resulting from the corresponding master equation are relatively easy to obtain; however, for practical applications one also needs to know the convergence to stationarity. The present work contributes to this direction, studying the transient dynamics in the discrete version of the model by two different approaches. The first method is based on mathematical induction by the recursive integration of the coupled differential equations for the discrete states. The second method transforms the coupled ordinary differential equation system into a partial differential equation for the generating function. We derive analytical results for some important, practically interesting cases and discuss the obtained results for the transient dynamics.

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