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.

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.

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.

A 3D Autonomous System with Infinitely Many Chaotic Attractors

2019 ◽  
Vol 29 (12) ◽  
pp. 1950166 ◽  
Author(s):  
Ting Yang ◽  
Qigui Yang
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Autonomous System ◽  
Three Dimensional ◽  
Chaotic Attractors ◽  
Double Zero ◽  
Finite Dimensional ◽  
Periodic Attractors ◽  
The Stability ◽  
Autonomous Dynamical System

Intuitively, a finite-dimensional autonomous system of ordinary differential equations can only generate finitely many chaotic attractors. Amazingly, however, this paper finds a three-dimensional autonomous dynamical system that can generate infinitely many chaotic attractors. Specifically, this system can generate infinitely many coexisting chaotic attractors and infinitely many coexisting periodic attractors in the following three cases: (i) no equilibria, (ii) only infinitely many nonhyperbolic double-zero equilibria, and (iii) both infinitely many hyperbolic saddles and nonhyperbolic pure-imaginary equilibria. By analyzing the stability of double-zero and pure-imaginary equilibria, it is shown that the classic Shil’nikov criteria fail in verifying the existence of chaos in the above three cases.

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 On the Dynamics of Coupled Parametrically Forced Oscillators

2008 ◽  
Author(s):  
Jeff Moehlis
Keyword(s):  
Dynamical System ◽  
Hopf Bifurcation ◽  
Periodic Orbit ◽  
Weak Coupling ◽  
Weak Coupling Limit ◽  
Stable Periodic Orbit ◽  
Weakly Coupled ◽  
Forced Oscillators ◽  
Stable Periodic ◽  
Autonomous Dynamical System

It is well known that an autonomous dynamical system can have a stable periodic orbit, arising for example through a Hopf bifurcation. When a collection of such oscillators is coupled together, the system can display a number of phase-locked solutions which can be understood in the weak coupling limit by using a phase model. It is also well known that a stable periodic orbit can be found for a parametrically forced dynamical system, with the phase of the periodic orbit being locked to the forcing. Here we discuss the periodic solutions which occur for a collection of such parametrically forced oscillators that are weakly coupled together.

scholarly journals A physical memristor based Muthuswamy–Chua–Ginoux system

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Jean-Marc Ginoux ◽  
Bharathwaj Muthuswamy ◽  
Riccardo Meucci ◽  
Stefano Euzzor ◽  
Angelo Di Garbo ◽  
...  
Keyword(s):  
Dynamical System ◽  
Chaotic System ◽  
Mathematical Analysis ◽  
Three Dimensional ◽  
Memristive Device ◽  
Nonlinear Resistor ◽  
Numerical Investigations ◽  
Double Spiral ◽  
Torus Breakdown ◽  
Autonomous Dynamical System

Abstract In 1976, Leon Chua showed that a thermistor can be modeled as a memristive device. Starting from this statement we designed a circuit that has four circuit elements: a linear passive inductor, a linear passive capacitor, a nonlinear resistor and a thermistor, that is, a nonlinear “locally active” memristor. Thus, the purpose of this work was to use a physical memristor, the thermistor, in a Muthuswamy–Chua chaotic system (circuit) instead of memristor emulators. Such circuit has been modeled by a new three-dimensional autonomous dynamical system exhibiting very particular properties such as the transition from torus breakdown to chaos. Then, mathematical analysis and detailed numerical investigations have enabled to establish that such a transition corresponds to the so-called route to Shilnikov spiral chaos but gives rise to a “double spiral attractor”.

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.

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