ABRUPT DIMENSION CHANGES AT BASIN BOUNDARY METAMORPHOSES

1992 ◽  
Vol 02 (03) ◽  
pp. 533-541 ◽  
Author(s):  
BAE-SIG PARK ◽  
CELSO GREBOGI ◽  
YING-CHENG LAI
Keyword(s):  
Dynamical Systems ◽  
Numerical Experiments ◽  
System Parameter ◽  
Analytic Calculation ◽  
Two Dimensional ◽  
Qualitative Model ◽  
One Dimensional ◽  
Basin Boundary ◽  
Chaotic Dynamical Systems ◽  
Basin Boundaries

Basin boundaries in chaotic dynamical systems can be either smooth or fractal. As a system parameter changes, the structure of the basin boundary also changes. In particular, the dimension of the basin boundary changes continuously except when a basin boundary metamorphosis occurs, at which it can change abruptly. We present numerical experiments to demonstrate such sudden dimension changes. We have also used a one-dimensional analytic calculation and a two-dimensional qualitative model to explain such changes.

scholarly journals Some numerical experiments on firn ventilation with heat transfer

1993 ◽  
Vol 18 ◽  
pp. 161-165 ◽  
Author(s):  
M.R. Albert
Keyword(s):  
Heat Transfer ◽  
Surface Pressure ◽  
Thermal Effects ◽  
Numerical Experiments ◽  
Temperature Gradients ◽  
Pressure Amplitude ◽  
Two Dimensional ◽  
One Dimensional ◽  
Thermal Signature ◽  
Spatially Varying

Preliminary estimates of the thermal signature of ventilation in polar firn are obtained from two-dimensional numerical calculations. The simulations show that spatially varying surface pressure can induce airflow velocities of 10−5m s−1at 1.5 m depth in uniform firn, and higher velocities closer to the surface. The two-dimensional heat-transfer results generally agree with our earlier one-dimensional conclusions that the thermal effects of ventilation tend to decrease the temperature gradient in the top portions of the pack. Field observations of ventilation through temperature measurements are most likely to be observed when the firn temperature at depths on the order of 10 m is close to the air temperature, since steep temperature gradients can mask the thermal effects of ventilation. Preliminary indications are that, as long as surface-pressure amplitude is sufficient to move the air about in the top tens of centimeters in the snow, the resulting temperature profile during ventilation is fairly insensitive to the frequency of the surface-pressure forcing for pressure frequencies in the range 0.1–10.0 Hz.

scholarly journals One-dimensional and two-dimensional dynamics of cubic maps

2006 ◽  
Vol 2006 ◽  
pp. 1-13 ◽  
Author(s):  
Djellit Ilhem ◽  
Kara Amel
Keyword(s):  
Large Class ◽  
System Parameter ◽  
Two Dimensional ◽  
One Dimensional ◽  
Henon Maps ◽  
Hénon Maps ◽  
Starting Point ◽  
Good Starting Point ◽  
Polynomial Diffeomorphisms

We concentrate on the dynamics of one-dimensional and two-dimensional cubic maps, it describes how complex behaviors can possibly arise as a system parameter changes. This is a large class of diffeomorphisms which provide a good starting point for understanding polynomial diffeomorphisms with constant Jacobian and equivalent to a composition of generalized Hénon maps. Due to the theoretical and practical difficulties involved in the study, computers will presumably play a role in such efforts.

scholarly journals Dweiltime Analysis of Symmetry-Breaking Dynamical Systems

1995 ◽  
Vol 50 (12) ◽  
pp. 1117-1122 ◽  
Author(s):  
J. Vollmer ◽  
J. Peinke ◽  
A. Okniński
Keyword(s):  
Dynamical Systems ◽  
Symmetry Breaking ◽  
Dwell Time ◽  
Initial Conditions ◽  
Dissipative Systems ◽  
Time Analysis ◽  
Chaotic Scattering ◽  
One Dimensional ◽  
Basin Boundary ◽  
Sensitive Dependence

Abstract Dweiltime analysis is known to characterize saddles giving rise to chaotic scattering. In the present paper it is used to characterize the dependence on initial conditions of the attractor approached by a trajectory in dissipative systems described by one-dimensional, noninvertible mappings which show symmetry breaking. There may be symmetry-related attractors in these systems, and which attractor is approached may depend sensitively on the initial conditions. Dwell-time analysis is useful in this context because it allows to visualize in another way the repellers on the basin boundary which cause this sensitive dependence.

THE TOPOLOGY OF BASIN BOUNDARIES IN A CLASS OF THREE-DIMENSIONAL DYNAMICAL SYSTEMS

1996 ◽  
Vol 06 (01) ◽  
pp. 161-167
Author(s):  
M. KLEMM ◽  
P. E. BECKMANN
Keyword(s):  
Dynamical Systems ◽  
Three Dimensional ◽  
Topological Invariants ◽  
Integration Algorithm ◽  
Topological Methods ◽  
Basin Boundary ◽  
New Methods ◽  
New Information ◽  
Basin Boundaries ◽  
Backward Integration

We will develop new methods to determine the topology of the basin boundary in a class of three-dimensional dynamical systems. One approach is to approximate the basin boundary by backward integration. Unfortunately, there are dynamical systems where it is hard to approximate the basin boundary by a numerical backward integration algorithm. We will introduce topological methods which will provide new information about the structure of the basin boundary. The topological invariants which we will use can be numerically computed.

Jacobi stability for dynamical systems of two-dimensional second-order differential equations and application to overhead crane system

2016 ◽  
Vol 13 (04) ◽  
pp. 1650045 ◽  
Author(s):  
Takahiro Yajima ◽  
Kazuhito Yamasaki
Keyword(s):  
Differential Equation ◽  
Dynamical Systems ◽  
Order Differential Equation ◽  
Geometric Theory ◽  
Transient State ◽  
Second Order ◽  
Two Dimensional ◽  
Overhead Crane ◽  
One Dimensional ◽  
Second Order Differential Equation

Geometric structures of dynamical systems are investigated based on a differential geometric method (Jacobi stability of KCC-theory). This study focuses on differences of Jacobi stability of two-dimensional second-order differential equation from that of one-dimensional second-order differential equation. One of different properties from a one-dimensional case is the Jacobi unstable condition given by eigenvalues of deviation curvature with different signs. Then, this geometric theory is applied to an overhead crane system as a two-dimensional dynamical system. It is shown a relationship between the Hopf bifurcation of linearized overhead crane and the Jacobi stability. Especially, the Jacobi stable trajectory is found for stable and unstable spirals of the two-dimensional linearized system. In case of the linearized overhead crane system, the Jacobi stable spiral approaches to the equilibrium point faster than the Jacobi unstable spiral. This means that the Jacobi stability is related to the resilience of deviated trajectory in the transient state. Moreover, for the nonlinear overhead crane system, the Jacobi stability for limit cycle changes stable and unstable over time.

CHAOTIC SYNCHRONIZATION OF A ONE-DIMENSIONAL ARRAY OF NONLINEAR ACTIVE SYSTEMS

1993 ◽  
Vol 03 (04) ◽  
pp. 1067-1074 ◽  
Author(s):  
V. PÉREZ-VILLAR ◽  
A. P. MUÑUZURI ◽  
V. PÉREZ-MUÑUZURI ◽  
L. O. CHUA
Keyword(s):  
Stability Analysis ◽  
Phase Space ◽  
Dynamical Systems ◽  
Chaotic Synchronization ◽  
Active Systems ◽  
One Dimensional ◽  
Chaotic Dynamical Systems ◽  
First Case ◽  
Theoretical Predictions

Linear stability analysis is used to study the synchronization of N coupled chaotic dynamical systems. It is found that the role of the coupling is always to stabilize the system, and then synchronize it. Computer simulations and experimental results of an array of Chua's circuits are carried out. Arrays of identical and slightly different oscillators are considered. In the first case, the oscillators synchronize and sync-phase, i.e., each one repeats exactly the same behavior as the rest of them. When the oscillators are not identical, they can also synchronize but not in phase with each other. The last situation is shown to form structures in the phase space of the dynamical variables. Due to the inevitable component tolerances (±5%), our experiments have so far confirmed our theoretical predictions only for an array of slightly different oscillators.

On the Fractal Structure of Basin Boundaries in Two-Dimensional Noninvertible Maps

2003 ◽  
Vol 13 (07) ◽  
pp. 1767-1785 ◽  
Author(s):  
A. Agliari ◽  
L. Gardini ◽  
C. Mira
Keyword(s):  
Fractal Structure ◽  
Real Plane ◽  
Two Dimensional ◽  
Numerical Examples ◽  
The Real ◽  
Basin Boundary ◽  
Fractal Basin Boundary ◽  
Noninvertible Maps ◽  
Analytical Tools ◽  
Basin Boundaries

In this paper we give an example of transition to fractal basin boundary in a two-dimensional map coming from the applicative context, in which the hard-fractal structure can be rigorously proved. That is, not only via numerical examples, although theoretically guided, as often occurs in maps coming from the applications, but also via analytical tools. The proposed example connects the two-dimensional maps of the real plane to the well-known complex map.

DERIVING CHAOTIC DYNAMICAL SYSTEMS FROM ENERGY FUNCTIONALS

2001 ◽  
Vol 01 (03) ◽  
pp. 377-388 ◽  
Author(s):  
PAUL BRACKEN ◽  
PAWEŁ GÓRA ◽  
ABRAHAM BOYARSKY
Keyword(s):  
Differential Equation ◽  
Dynamical Systems ◽  
Order Differential Equation ◽  
Lagrange Equation ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Energy Functionals ◽  
Chaotic Dynamical Systems ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

Simple one-dimensional chaotic dynamical systems are derived by optimizing energy functionals. The Euler–Lagrange equation yields a nonlinear second-order differential equation whose solution yields a 2–1 map which admits an absolutely continuous invariant measure. The solutions of the differential equation are studied.

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