SYNCHRONIZATION OF CHAOS AND THE TRANSITION TO WAVE TURBULENCE

2012 ◽  
Vol 22 (10) ◽  
pp. 1250234 ◽  
Author(s):  
R. L. VIANA ◽  
S. R. LOPES ◽  
J. D. SZEZECH ◽  
I. L. CALDAS
Keyword(s):  
Chaotic Dynamics ◽  
Spatial Dynamics ◽  
Wave Turbulence ◽  
Homogeneous State ◽  
Spatial Mode ◽  
Mode Excitation ◽  
Spatially Extended ◽  
Spatially Homogeneous

We investigated the transition to wave turbulence in a spatially extended three-wave interacting model, where a spatially homogeneous state undergoing chaotic dynamics undergoes spatial mode excitation. The transition to this weakly turbulent state can be regarded as the loss of synchronization of chaos of mode oscillators describing the spatial dynamics.

Curvature effects and radial homoclinic snaking

2021 ◽  
Author(s):  
Damià Gomila ◽  
Edgar Knobloch
Keyword(s):  
Scaling Law ◽  
Spatial Dynamics ◽  
Two Dimensions ◽  
Localized States ◽  
Homogeneous State ◽  
Curvature Effects ◽  
Localized Structures ◽  
Homoclinic Snaking ◽  
Curvature Driven ◽  
Axisymmetric Structures

Abstract In this work, we revisit some general results on the dynamics of circular fronts between homogeneous states and the formation of localized structures in two dimensions (2D). We show how the bifurcation diagram of axisymmetric structures localized in radius fits within the framework of collapsed homoclinic snaking. In 2D, owing to curvature effects, the collapse of the snaking structure follows a different scaling that is determined by the so-called nucleation radius. Moreover, in the case of fronts between two symmetry-related states, the precise point in parameter space to which radial snaking collapses is not a ‘Maxwell’ point but is determined by the curvature-driven dynamics only. In this case, the snaking collapses to a ‘zero surface tension’ point. Near this point, the breaking of symmetry between the homogeneous states tilts the snaking diagram. A different scaling law is found for the collapse of the snaking curve in each case. Curvature effects on axisymmetric localized states with internal structure are also discussed, as are cellular structures separated from a homogeneous state by a circular front. While some of these results are well understood in terms of curvature-driven dynamics and front interactions, a proper mathematical description in terms of homoclinic trajectories in a radial spatial dynamics description is lacking.

SPATIO-TEMPORAL PATTERNS IN A CHOLERA TRANSMISSION MODEL

2015 ◽  
Vol 23 (03) ◽  
pp. 471-484 ◽  
Author(s):  
A. K. MISRA ◽  
MILAN TIWARI ◽  
ANUPAMA SHARMA
Keyword(s):  
Spatial Patterns ◽  
Dimensional Space ◽  
Spatial Dynamics ◽  
Transmission Model ◽  
Cholera Epidemic ◽  
Temporal System ◽  
Spatially Extended ◽  
Variable Population ◽  
Spatio Temporal ◽  
Variable Population Size

Cholera has been a public health threat for centuries. Unlike the biological characteristics, relatively less effort has been paid to comprehend the spatial dynamics of this disease. Therefore, in this paper, we have proposed a cholera epidemic model for variable population size and studied the spatial patterns in two-dimensional space. First, we have performed the equilibrium and local stability analysis of steady states obtained for temporal system. Afterwards, the local and global stability behavior of the endemic steady state in a spatially extended setting has been investigated. The numerical simulations have been done to investigate the spatial patterns. They show that dynamics of the cholera epidemic varies with time and space.

The consequences of stochasticity for self-organized spatial dynamics, persistence and coexistence in spatially extended host-parasitoid communities

1996 ◽  
Vol 263 (1370) ◽  
pp. 625-631 ◽  
Keyword(s):  
Spatial Dynamics ◽  
The Other ◽  
Spatially Explicit ◽  
Parasitoid Species ◽  
Spatially Explicit Model ◽  
Self Organized ◽  
Spatially Extended ◽  
Spatio Temporal ◽  
Parameter Values ◽  
Short Range Correlations

We investigate the behaviour of a spatially explicit model of the interaction between two parasitoid species and their common host. One parasitoid species is able to move between host subpopulations at a faster rate than the other parasitoid species which has a higher attack rate. Without space, the model has no equilibrium. With the addition of space, however, Comins & Hassell (1996) have shown that persistence of at least one parasitoid species is generally observed and coexistence of the two parasitoid species can be obtained over a range of parameter values. They observe that this persistence is accompanied by spatial segregation of the competing species within ‘self-organized’ spiral patterns. Here, we investigate the effects of adding various forms of temporal and spatio-temporal stochasticity to the model, and demonstrate that low-to-moderate levels of noise generally inhibit the system from forming clearly defined spirals. Despite this, there are still strong short-range correlations in species densities and this spatial heterogeneity is sufficient to allow persistence and coexistence of competitors. The addition of noise acts to increase the parameter range where the more mobile parasitoid is excluded by the other and decreases the range where the more mobile parasitoid excludes its competitor. Even if the perturbation is strong, for example, with all individuals in a randomly selected 10% of sites being eliminated at each generation, then persistence still occurs and coexistence can be achieved over a suitable range of parameters. Again, the competitive advantage of the more mobile parasitoid is reduced in the presence of this perturbation.

scholarly journals Rationals, periodicity and chaos: A Pythagorean view and a conjecture into socio-spatial dynamics

2000 ◽  
Vol 4 (2) ◽  
pp. 133-143
Author(s):  
Dimitrios S. Dendrinos
Keyword(s):  
Number Theory ◽  
Chaotic Motion ◽  
Chaotic Dynamics ◽  
Social Systems ◽  
Spatial Dynamics ◽  
Chaotic Sequence ◽  
Systems Dynamics ◽  
Specific Point ◽  
Chaotic Motions ◽  
Periodic Cycles

Deep in the fascinating world of numbers there still might lurk useful insights into the processes of the socio-spatial world. A rich section of the world of numbers is of course Number Theory and its pantheon of findings, a part of which is revisited here.It is suggested in this note that a smooth sequence of seemingly random periodic cycles hides the absence of chaotic dynamics in the sequence. Put differently, a seemingly chaotic sequence of periodic cycles, no matter the bandwidth, implies absence of chaotic motion at any point in the sequence; and conversely, the presence of chaotic motion at any specific point in the sequence implies smooth sequence of periodic cycles at any point in the sequence prior to the onset of quasi periodic or chaotic motions.To make this conjecture, the paper draws material from the well known property of rational numbers in Number Theory, namely that the division of unity by any integer will always produce a sequence of decimals in some form of periodicity. The conjecture is taken in a liberally interpreted “Pythagorean type” context, whereby a general principle is suggested to be present in all natural or social systems dynamics. Thus, the paper's subtitle.

CHARACTERIZATION OF CHAOTIC ATTRACTORS IN EXTENDED SYSTEMS

1996 ◽  
Vol 06 (07) ◽  
pp. 1375-1382 ◽  
Author(s):  
CHIEN-CHONG CHEN ◽  
CHYI HWANG ◽  
EDUARDO E. WOLF ◽  
HSUEH-CHIA CHANG
Keyword(s):  
Chaotic Dynamics ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Chaotic Attractors ◽  
Extended System ◽  
Embedding Method ◽  
Spatial Point ◽  
Spatially Distributed ◽  
Spatially Extended

It is shown that the traditional delayed embedding method, which is often applied to measurements at a single spatial point, could only account for partial chaotic dynamics of an extended system. To catch the whole dynamics of an extended system, we apply in this letter the Karhunen-Loeve (K-L) procedure to construct a series of tangent maps from spatially distributed measurements. The tangent maps obtained are then used for computing certain ergodic invariants. The procedure is illustrated with the experimental data obtained from a catalytic wafer, which is a spatially extended system exhibiting complex dynamics. The computed results show clearly that the K-L procedure is more suitable than the delayed embedding method for deciphering the full chaotic behavior of an extended system.

Bubbling transition to spatial mode excitation in an extended dynamical system

2009 ◽  
Vol 238 (5) ◽  
pp. 516-525 ◽  
Author(s):  
J.D. Szezech ◽  
S.R. Lopes ◽  
R.L. Viana ◽  
I.L. Caldas
Keyword(s):  
Dynamical System ◽  
Spatial Mode ◽  
Mode Excitation
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