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.

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.

scholarly journals Dynamics of the Modified n-Degree Lorenz System

2019 ◽  
Vol 4 (2) ◽  
pp. 315-330 ◽  
Author(s):  
Sk. Sarif Hassan ◽  
Moole Parameswar Reddy ◽  
Ranjeet Kumar Rout
Keyword(s):  
Dynamical Systems ◽  
Fixed Points ◽  
Lyapunov Exponents ◽  
Limit Cycle ◽  
Chaotic Dynamics ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Lorenz Model ◽  
Periodic Behavior ◽  
Chaotic Model

AbstractThe Lorenz model is one of the most studied dynamical systems. Chaotic dynamics of several modified models of the classical Lorenz system are studied. In this article, a new chaotic model is introduced and studied computationally. By finding the fixed points, the eigenvalues of the Jacobian, and the Lyapunov exponents. Transition from convergence behavior to the periodic behavior (limit cycle) are observed by varying the degree of the system. Also transiting from periodic behavior to the chaotic behavior are seen by changing the degree of the system.

CHAOS IN A NEAR-INTEGRABLE HAMILTONIAN LATTICE

2002 ◽  
Vol 12 (08) ◽  
pp. 1743-1754 ◽  
Author(s):  
VASSILIOS M. ROTHOS ◽  
CHRIS ANTONOPOULOS ◽  
LAMBROS DROSSOS
Keyword(s):  
Lyapunov Exponents ◽  
Numerical Experiments ◽  
Chaotic Dynamics ◽  
Large Scale ◽  
Hamiltonian Structure ◽  
Chaotic Behavior ◽  
Original System ◽  
Homoclinic Orbits ◽  
Hamiltonian Lattice ◽  
Whiskered Tori

We study the chaotic dynamics of a near-integrable Hamiltonian Ablowitz–Ladik lattice, which is N + 2-dimensional if N is even (N + 1, if N is odd) and possesses, for all N, a circle of unstable equilibria at ε = 0, whose homoclinic orbits are shown to persist for ε ≠ 0 on whiskered tori. The persistence of homoclinic orbits is established through Mel'nikov conditions, directly from the Hamiltonian structure of the equations. Numerical experiments which combine space portraits and Lyapunov exponents are performed for the perturbed Ablowitz–Ladik lattice and large scale chaotic behavior is observed in the vicinity of the circle of unstable equilibria in the ε = 0 case. We conjecture that this large scale chaos is due to the occurrence of saddle-center type fixed points in a perturbed 1 d.o.f Hamiltonian to which the original system can be reduced for all N. As ε > 0 increases, the transient character of this chaotic behavior becomes apparent as the positive Lyapunov exponents steadily increase and the orbits escape to infinity.

CONSTANT-PHASE-INDUCED CONTROL OF CHAOTIC CHARGED PARTICLES IN THE FIELD OF A WAVE PACKET

2013 ◽  
Vol 23 (06) ◽  
pp. 1330022
Author(s):  
RICARDO CHACÓN
Keyword(s):  
Charged Particle ◽  
Wave Packet ◽  
Finite Number ◽  
Lyapunov Exponents ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Charged Particles ◽  
Melnikov Method ◽  
Wide Range ◽  
Range Of Values

It is shown that the dissipative chaotic dynamics of a charged particle in the field of a wave packet with an arbitrary but finite number of harmonics can be reliably suppressed by judiciously varying the constant phase of the main harmonic, ϕ0, while keeping null the corresponding constant phases of the remaining harmonics. The dependence of the chaotic threshold on the wave packet parameters is predicted theoretically (Melnikov method) and confirmed numerically (Lyapunov exponents). In particular, it is shown that ϕ0 is effective at suppressing the chaotic behavior existing when ϕ0 = 0 over a wide range of values of the wave packet width, while the remaining parameters are kept constant.

Stability analysis of a fractional-order cancer model with chaotic dynamics

2021 ◽  
pp. 2150046
Author(s):  
Parvaiz Ahmad Naik ◽  
Jian Zu ◽  
Mehraj-ud-din Naik
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Cancer Model ◽  
Effector Cells ◽  
Proposed Model ◽  
Uniqueness Of The Solution ◽  
Theoretical Results

In this paper, we develop a three-dimensional fractional-order cancer model. The proposed model involves the interaction among tumor cells, healthy tissue cells and activated effector cells. The detailed analysis of the equilibrium points is studied. Also, the existence and uniqueness of the solution are investigated. The fractional derivative is considered in the Caputo sense. Numerical simulations are performed to illustrate the effectiveness of the obtained theoretical results. The outcome of the study reveals that the order of the fractional derivative has a significant effect on the dynamic process. Further, the calculated Lyapunov exponents give the existence of chaotic behavior of the proposed model. Also, it is observed from the obtained results that decrease in fractional-order [Formula: see text] increases the chaotic behavior of the model.

scholarly journals The Dynamics of a Cubic Nonlinear System with No Equilibrium Point

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
J. O. Maaita ◽  
Ch. K. Volos ◽  
I. M. Kyprianidis ◽  
I. N. Stouboulos
Keyword(s):  
Nonlinear System ◽  
Equilibrium Point ◽  
Lyapunov Exponents ◽  
Electronic Circuit ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Circuit Modeling ◽  
Cubic Nonlinearity ◽  
Poincaré Maps

We study the dynamics of a three-dimensional nonlinear system with cubic nonlinearity and no equilibrium points with the use of Poincaré maps, Lyapunov Exponents, and bifurcations diagrams. The system has rich dynamics: chaotic behavior, regular orbits, and 3-tori periodicity. Finally, the proposed system is also reported to verify electronic circuit modeling feasibility.

BIFURCATION SCENARIO OF A THREE-DIMENSIONAL VAN DER POL OSCILLATOR

1993 ◽  
Vol 03 (02) ◽  
pp. 399-404 ◽  
Author(s):  
T. SÜNNER ◽  
H. SAUERMANN
Keyword(s):  
Bifurcation Diagram ◽  
Bifurcation Analysis ◽  
Chaotic Behavior ◽  
Three Dimensional ◽  
Global Bifurcation ◽  
Dynamical Behaviour ◽  
Van Der Pol Oscillator ◽  
Two Dimensional ◽  
Van Der Pol ◽  
Dimensional Models

Nonlinear self-excited oscillations are usually investigated for two-dimensional models. We extend the simplest and best known of these models, the van der Pol oscillator, to a three-dimensional one and study its dynamical behaviour by methods of bifurcation analysis. We find cusps and other local codimension 2 bifurcations. A homoclinic (i.e. global) bifurcation plays an important role in the bifurcation diagram. Finally it is demonstrated that chaos sets in. Thus the system belongs to the few three-dimensional autonomous ones modelling physical situations which lead to chaotic behavior.

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.

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