Three-Dimensional Torus Breakdown and Chaos With Two Zero Lyapunov Exponents in Coupled Radio-Physical Generators

2020 ◽  
Vol 15 (11) ◽  
Author(s):  
Nataliya V. Stankevich ◽  
Natalya A. Shchegoleva ◽  
Igor R. Sataev ◽  
Alexander P. Kuznetsov
Keyword(s):  
Lyapunov Exponents ◽  
Chaotic Dynamics ◽  
Three Dimensional ◽  
Parameter Plane ◽  
Poincare Section ◽  
Dimensional Torus ◽  
Typical Structure ◽  
Periodic Oscillations ◽  
Torus Breakdown ◽  
Periodic Dynamics

Abstract Using an example a system of two coupled generators of quasi-periodic oscillations, we study the occurrence of chaotic dynamics with one positive, two zero, and several negative Lyapunov exponents. It is shown that such dynamic arises as a result of a sequence of bifurcations of two-frequency torus doubling and involves saddle tori occurring at their doublings. This transition is associated with typical structure of parameter plane, like cross-road area and shrimp-shaped structures, based on the two-frequency quasi-periodic dynamics. Using double Poincaré section, we have shown destruction of three-frequency torus.

On the integrability of intermediate distributions for Anosov diffeomorphisms

1995 ◽  
Vol 15 (2) ◽  
pp. 317-331 ◽  
Author(s):  
M. Jiang ◽  
Ya B. Pesin ◽  
R. de la Llave
Keyword(s):  
Finite Number ◽  
Lyapunov Exponents ◽  
Periodic Orbits ◽  
Three Dimensional ◽  
Dimensional Torus ◽  
Anosov Diffeomorphism ◽  
Anosov Diffeomorphisms ◽  
Stable Foliation

AbstractWe study the integrability of intermediate distributions for Anosov diffeomorphisms and provide an example of a C∞-Anosov diffeomorphism on a three-dimensional torus whose intermediate stable foliation has leaves that admit only a finite number of derivatives. We also show that this phenomenon is quite abundant. In dimension four or higher this can happen even if the Lyapunov exponents at periodic orbits are constant.

scholarly journals Spatiotemporal dynamics in a diffusive Holling-Tanner model near codimension-two bifurcations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Daifeng Duan ◽  
Ben Niu ◽  
Junjie Wei
Keyword(s):  
Generation Time ◽  
Three Dimensional ◽  
Spatiotemporal Patterns ◽  
Hopf Bifurcations ◽  
Dimensional Torus ◽  
Periodic Oscillations ◽  
One Dimensional ◽  
Codimension Two ◽  
Spatially Inhomogeneous ◽  
Delayed System

<p style='text-indent:20px;'>We investigate spatiotemporal patterns near the Turing-Hopf and double Hopf bifurcations in a diffusive Holling-Tanner model on a one- dimensional spatial domain. Local and global stability of the positive constant steady state for the non-delayed system is studied. Introducing the generation time delay in prey growth, we discuss the existence of Turing-Hopf and double Hopf bifurcations and give the explicit dynamical classification near these bifurcation points. Finally, we obtain the complicated dynamics, including periodic oscillations, quasi-periodic oscillations on a three-dimensional torus, the coexistence of two stable nonconstant steady states, the coexistence of two spatially inhomogeneous periodic solutions, and strange attractors.</p>

CHAOS IN A THREE-DIMENSIONAL CANCER MODEL

2010 ◽  
Vol 20 (01) ◽  
pp. 71-79 ◽  
Author(s):  
MEHMET ITIK ◽  
STEPHEN P. BANKS
Keyword(s):  
Immune System ◽  
Lyapunov Exponents ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Three Dimensional ◽  
Dynamical Model ◽  
Parameter Range ◽  
Cancer Growth ◽  
Cancer Model ◽  
Lyapunov Dimension

In this study, we develop a new dynamical model of cancer growth, which includes the interactions between tumour cells, healthy tissue cells, and activated immune system cells, clearly leading to chaotic behavior. We explain the biological relevance of our model and the ways in which it differs from the existing ones. We perform equilibria analysis, indicate the conditions where chaotic dynamics can be observed, and show rigorously the existence of chaos by calculating the Lyapunov exponents and the Lyapunov dimension of the system. Moreover, we demonstrate that Shilnikov's theorem is valid in the parameter range of interest.

scholarly journals A computational framework for finding parameter sets associated with chaotic dynamics

2021 ◽  
pp. 1-11
Author(s):  
S. Koshy-Chenthittayil ◽  
E. Dimitrova ◽  
E.W. Jenkins ◽  
B.C. Dean
Keyword(s):  
Experimental Data ◽  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Parameter Space ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Computational Framework ◽  
Independent Variables ◽  
Systems Model ◽  
Small Support

Many biological ecosystems exhibit chaotic behavior, demonstrated either analytically using parameter choices in an associated dynamical systems model or empirically through analysis of experimental data. In this paper, we use existing software tools (COPASI, R) to explore dynamical systems and uncover regions with positive Lyapunov exponents where thus chaos exists. We evaluate the ability of the software’s optimization algorithms to find these positive values with several dynamical systems used to model biological populations. The algorithms have been able to identify parameter sets which lead to positive Lyapunov exponents, even when those exponents lie in regions with small support. For one of the examined systems, we observed that positive Lyapunov exponents were not uncovered when executing a search over the parameter space with small spacings between values of the independent variables.

scholarly journals A New Test for Chaotic Dynamics Using Lyapunov Exponents

2003 ◽  
Author(s):  
Fernando Fernández Rodríguez ◽  
Simón Javier Sosvilla Rivero ◽  
Julián Andrada Félix
Keyword(s):  
Lyapunov Exponents ◽  
Chaotic Dynamics

CHAOTIC DYNAMICS OF A SIMPLE PARAMETRICALLY DRIVEN DISSIPATIVE CIRCUIT

2011 ◽  
Vol 21 (07) ◽  
pp. 1927-1933 ◽  
Author(s):  
P. PHILOMINATHAN ◽  
M. SANTHIAH ◽  
I. RAJA MOHAMED ◽  
K. MURALI ◽  
S. RAJASEKAR
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Exponents ◽  
Chaotic Dynamics ◽  
Period Doubling ◽  
Classic Period ◽  
Control Signal ◽  
Period Doubling Bifurcation ◽  
Chaotic Circuit ◽  
Route To Chaos ◽  
Parameter Values

We introduce a simple parametrically driven dissipative second-order chaotic circuit. In this circuit, one of the circuit parameters is varied by an external periodic control signal. Thus by tuning the parameter values of this circuit, classic period-doubling bifurcation route to chaos is found to occur. The experimentally observed phenomena is further validated through corresponding numerical simulation of the circuit equations. The periodic and chaotic dynamics of this model is further characterized by computing Lyapunov exponents.

scholarly journals Simulation Investigation on the Flow and Mixing in Ducts of the Rotary Energy Recovery Device

Geofluids ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Kai Liu ◽  
Lixing Zheng
Keyword(s):  
Fluid Dynamics ◽  
Energy Recovery ◽  
Swirling Flow ◽  
Three Dimensional ◽  
Working Process ◽  
Mixing Rate ◽  
Typical Structure ◽  
Mixing Performance ◽  
Swirl Intensity ◽  
High Turbulence

The rotary energy recovery device (RERD) is widely equipped in desalination to reduce the system energy consumption. In this study, the fluid dynamics and mixing performance of a typical structure RERD and a visualization apparatus of a RERD (V-RERD) had been compared by simulation. The effects of rotating components on fluid dynamics and mixing had been researched. Simulation results indicated that a swirling flow can be observed from flow fields in the device duct. In the RERD case, the swirling flow changed its rotating direction in the center of the duct, while in the V-RERD case, its rotating direction was unchanged. Then, a swirling number Sn was applied to characterize the degree of swirl intensity, and its formation mechanism in RERD had been discussed. In addition, the Q criterion was adopted to visualize the three-dimensional flow structures and identify vortex structures in the duct. The evolution of vortices in the working process had been investigated. It was found that vortices significantly affected the mixing performance, and the detached vortex could lead to high turbulence and mixing in the duct. Suppressing the vortex shedding may reduce the flow turbulence and gain a lower volumetric mixing rate.

ROBUSTNESS OF ADAPTATION IN CONTROLLED SELF-ADJUSTING CHAOTIC SYSTEMS

2002 ◽  
Vol 02 (04) ◽  
pp. L285-L292 ◽  
Author(s):  
PAUL MELBY ◽  
NICHOLAS WEBER ◽  
ALFRED HÜBLER
Keyword(s):  
Chaotic Dynamics ◽  
Chaotic Systems ◽  
Logistic Map ◽  
Loop Control ◽  
Edge Of Chaos ◽  
Target Value ◽  
Chua Circuit ◽  
Modified Logistic Map ◽  
Periodic Dynamics ◽  
Strong Adaptation

It was recently shown that self-adjusting systems adapt to the edge of chaos. We study the robustness of that adaptation with respect to a controlling force. We first use numerical simulations in a modified logistic map. With these, we find that, if the controlling force has a target value of the parameter that leads to periodic dynamics, the control is successful, even for very small controlling forces. We also find, however, that if the target value for the parameter leads to chaotic dynamics, the parameter resists the control and adaptation to the edge of chaos is still observed. When the controlling force is very strong, adaptation to the edge of chaos is weaker, but still present in the system. We also perform experiments with a self-adjusting Chua circuit and find the same behavior. We quantify these results with a measurement of the robustness of the adaptation as a function of the strength of the controlling force. The control used can be expressed either as a parametric control or as an additive, closed-loop control.

Sign in / Sign up

Export Citation Format

Share Document