DYNAMICS AT INFINITY OF A CUBIC CHUA'S SYSTEM

2011 ◽  
Vol 21 (01) ◽  
pp. 333-340 ◽  
Author(s):  
MARCELO MESSIAS
Keyword(s):  
Dynamical System ◽  
Vector Field ◽  
Numerical Study ◽  
Three Dimensional ◽  
Polynomial Vector ◽  
Cubic Nonlinearity ◽  
Poincaré Compactification ◽  
Global Study ◽  
Dimensional Dynamical System ◽  
Parameter Values

We use the Poincaré compactification for a polynomial vector field in ℝ3 to study the dynamics near and at infinity of the classical Chua's system with a cubic nonlinearity. We give a complete description of the phase portrait of this system at infinity, which is identified with the sphere 𝕊2 in ℝ3 after compactification, and perform a numerical study on how the solutions reach infinity, depending on the parameter values. With this global study we intend to give a contribution in the understanding of this well known and extensively studied complex three-dimensional dynamical system.

Viewing tropical forest succession as a three-dimensional dynamical system

2015 ◽  
Vol 9 (2) ◽  
pp. 163-172 ◽  
Author(s):  
Wirong Chanthorn ◽  
Yingluck Ratanapongsai ◽  
Warren Y. Brockelman ◽  
Michael A. Allen ◽  
Charly Favier ◽  
...  
Keyword(s):  
Dynamical System ◽  
Tropical Forest ◽  
Forest Succession ◽  
Three Dimensional ◽  
Dimensional Dynamical System

scholarly journals Asymptotic (statistical) periodicity in two-dimensional maps

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Fumihiko Nakamura ◽  
Michael C. Mackey
Keyword(s):  
Dynamical System ◽  
Bounded Variation ◽  
High Dimensional ◽  
Sufficient Condition ◽  
Two Dimensional ◽  
Function Of Bounded Variation ◽  
Dimensional Dynamical System ◽  
Asymptotic Periodicity ◽  
Parameter Values

<p style='text-indent:20px;'>In this paper we give a new sufficient condition for the existence of asymptotic periodicity of Frobenius–Perron operators corresponding to two–dimensional maps. Asymptotic periodicity for strictly expanding systems, that is, all eigenvalues of the system are greater than one, in a high-dimensional dynamical system was already known. Our new result enables one to deal with systems having an eigenvalue smaller than one. The key idea for the proof is to use a function of bounded variation defined by line integration. Finally, we introduce a new two-dimensional dynamical system numerically exhibiting asymptotic periodicity with different periods depending on parameter values, and discuss the application of our theorem to the example.</p>

scholarly journals On the nonsmooth dynamics of towed wheels

Meccanica ◽  
2020 ◽  
Vol 55 (12) ◽  
pp. 2523-2540 ◽  
Author(s):  
Mate Antali ◽  
Gabor Stepan
Keyword(s):  
Dynamical System ◽  
Dry Friction ◽  
Three Dimensional ◽  
Nonsmooth Dynamics ◽  
Filippov Systems ◽  
Dimensional Dynamical System ◽  
Creep Models

AbstractIn this paper, a nonsmooth model of towed wheels is analysed; this mechanism can be a part of different kind of vehicles. We focus on the transitions between slipping and rolling in the presence of dry friction. The model leads to a three-dimensional dynamical system with a codimension-2 discontinuity. The systems can be analysed by means of the tools of extended Filippov systems. The essence of the calculation is to find the so-called limit directions, which show the possible directions of slipping-rolling transitions and their properties. By this method, four different scenarios are found. The results are compared to those from the creep models.

A Four-Leaf Chaotic Attractor of a Three-Dimensional Dynamical System

2015 ◽  
Vol 25 (02) ◽  
pp. 1530003 ◽  
Author(s):  
Tomoyuki Miyaji ◽  
Hisashi Okamoto ◽  
Alex D. D. Craik
Keyword(s):  
Dynamical System ◽  
Three Dimensional ◽  
Period Doubling ◽  
Numerical Verification ◽  
Unstable Periodic Orbits ◽  
Verification Method ◽  
Stable Periodic ◽  
Approach Infinity ◽  
Dimensional Dynamical System ◽  
Autonomous Dynamical System

A three-dimensional autonomous dynamical system proposed by Pehlivan is untypical in simultaneously possessing both unbounded and chaotic solutions. Here, this topic is studied in some depth, both numerically and analytically. We find, by standard methods, that four-leaf chaotic orbits result from a period-doubling cascade; we identify unstable fixed points and both stable and unstable periodic orbits; and we examine how initial data determines whether orbits approach infinity or a stable periodic orbit. Further, we describe and apply a strict numerical verification method that rigorously proves the existence of sequences of period doublings.

Integrability and Dynamics of Quadratic Three-Dimensional Differential Systems Having an Invariant Paraboloid

2016 ◽  
Vol 26 (08) ◽  
pp. 1650134 ◽  
Author(s):  
Marcelo Messias ◽  
Alisson C. Reinol
Keyword(s):  
Three Dimensional ◽  
Algebraic Surfaces ◽  
Quadratic Polynomial ◽  
Poincaré Compactification ◽  
Differential Systems ◽  
Elliptic Paraboloid ◽  
Polynomial Differential Systems ◽  
Invariant Algebraic Surfaces ◽  
Exponential Factors ◽  
Parameter Values

Invariant algebraic surfaces are commonly observed in differential systems arising in mathematical modeling of natural phenomena. In this paper, we study the integrability and dynamics of quadratic polynomial differential systems defined in [Formula: see text] having an elliptic paraboloid as an invariant algebraic surface. We obtain the normal form for these kind of systems and, by using the invariant paraboloid, we prove the existence of first integrals, exponential factors, Darboux invariants and inverse Jacobi multipliers, for suitable choices of parameter values. We characterize all the possible configurations of invariant parallels and invariant meridians on the invariant paraboloid and give necessary conditions for the invariant parallel to be a limit cycle and for the invariant meridian to have two orbits heteroclinic to a point at infinity. We also study the dynamics of a particular class of the quadratic polynomial differential systems having an invariant paraboloid, giving information about localization and local stability of finite singular points and, by using the Poincaré compactification, we study their dynamics on the Poincaré sphere (at infinity). Finally, we study the well-known Rabinovich system in the case of invariant paraboloids, performing a detailed study of its dynamics restricted to these invariant algebraic surfaces.

scholarly journals Persistent structures in a three-dimensional dynamical system with flowing and non-flowing regions

2018 ◽  
Vol 9 (1) ◽  
Author(s):  
Zafir Zaman ◽  
Mengqi Yu ◽  
Paul P. Park ◽  
Julio M. Ottino ◽  
Richard M. Lueptow ◽  
...  
Keyword(s):  
Dynamical System ◽  
Three Dimensional ◽  
Dimensional Dynamical System

scholarly journals Almost oscillatory three-dimensional dynamical system

2012 ◽  
Vol 2012 (1) ◽  
Author(s):  
Elvan Akin-Bohner ◽  
Zuzana Došlá ◽  
Bonita Lawrence
Keyword(s):  
Dynamical System ◽  
Three Dimensional ◽  
Dimensional Dynamical System

DYNAMICS OF A SIMPLE ENDOGENOUS GROWTH MODEL WITH FINANCIAL INTERMEDIATION

2019 ◽  
pp. 1-15
Author(s):  
Dominika Byrska ◽  
Adam Krawiec ◽  
Marek Szydłowski
Keyword(s):  
Dynamical System ◽  
Economic Growth ◽  
Invariant Manifolds ◽  
Financial Intermediation ◽  
Three Dimensional ◽  
Mathematical Methods ◽  
Endogenous Growth Model ◽  
Dimensional Dynamical System ◽  
Insight Into ◽  
Autonomous Dynamical System

We study an impact of the financial intermediation on economic growth. We assume the simple model of the economic growth in the form of an autonomous dynamical system with a financial sector represented by banks and real sector represented by households and firms. We assume that financial intermediation services are described by financial intermediation technology which is a function depending on the share of labor employed by banks. Investments realized by firms depend not only on savings accumulated by banks but also on financial intermediation technology. We obtain a three-dimensional dynamical system and analyze the existence of a saddle equilibrium in the growth process associated with financial intermediation. Using mathematical methods of dynamical systems, we analyze growth paths, and we study the stationary states of the system and their stability. We found that equilibrium is reached only by trajectories located on two submanifolds. The resulting analysis provides an insight into the saddle solution with a stable incoming separatrix lying on one of the invariant manifolds.

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