Qualitative analysis of the Rössler equations: bifurcations of limit cycles and chaotic attractors

DOI10.1016/j.physd.2009.03.010zbMath1173.37049OpenAlexW2094871671MaRDI QIDQ2389361

Fernando Blesa, Roberto Barrio, Sergio E. Serrano

Publication date: 15 July 2009

Published in: Physica D (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.physd.2009.03.010

zbMATH Keywords

bifurcationschaotic attractorsroutes to chaoschaos indicatorsRössler equationsOFLI2

Mathematics Subject Classification ID

Bifurcation theory for ordinary differential equations (34C23) Dynamics induced by flows and semiflows (37C10) Strange attractors, chaotic dynamics of systems with hyperbolic behavior (37D45) Bifurcations of limit cycles and periodic orbits in dynamical systems (37G15) Dynamical aspects of attractors and their bifurcations (37G35)

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Uses Software

• MATCONT
• AUTO
• TIDES

Cites Work

• Spurious structures in chaos indicators maps
• A global qualitative view of bifurcations and dynamics in the Rössler system
• Inequivalent topologies of chaos in simple equations
• VSVO formulation of the Taylor method for the numerical solution of ODEs
• A three-parametric study of the Lorenz model
• Determining Lyapunov exponents from a time series
• What can we learn from homoclinic orbits in chaotic dynamics ?
• Local and global behavior near homoclinic orbits
• Bifurcation phenomena near homoclinic systems: A two-parameter analysis
• Phase space structure of multi-dimensional systems by means of the mean exponential growth factor of nearby orbits
• Periodic solutions to the Rössler system
• Knotted periodic orbits in dynamical systems. I: Lorenz's equations
• On the relationship between fast Lyapunov indicator and periodic orbits for continuous flows
• 3-dimensional Hopf bifurcation via averaging theory
• An equation for continuous chaos
• Localization of periodic orbits of the Rössler system under variation of its parameters
• Sensitivity tools vs. Poincaré sections
• A new test for chaos in deterministic systems
• Topological analysis of chaotic dynamical systems
• Algorithm 924
• PAINTING CHAOS: A GALLERY OF SENSITIVITY PLOTS OF CLASSICAL PROBLEMS
• COUNTING LOW-PERIOD CYCLES FOR FLOWS
• FORMAL AND ANALYTIC INTEGRABILITY OF THE ROSSLER SYSTEM
• CHARACTERIZATION OF THE RÖSSLER SYSTEM IN PARAMETER SPACE
• RESONANCES OF PERIODIC ORBITS IN RÖSSLER SYSTEM IN PRESENCE OF A TRIPLE-ZERO BIFURCATION
• Computer assisted proof of chaos in the Rössler equations and in the Hénon map
• NUMERICAL DETECTION AND CONTINUATION OF CODIMENSION-TWO HOMOCLINIC BIFURCATIONS
• BIFURCATIONS IN THE COLPITTS OSCILLATOR: FROM THEORY TO PRACTICE
• NEW RESULTS ABOUT ROUTE TO CHAOS IN ROSSLER SYSTEM
• QUALITATIVE RESONANCE OF SHIL'NIKOV-LIKE STRANGE ATTRACTORS, PART II: MATHEMATICAL ANALYSIS
• Application of the 0-1 Test for Chaos to Experimental Data
• MATCONT
• Sensitivity Analysis of ODES/DAES Using the Taylor Series Method
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