Necessary and sufficient conditions of the existence of homoclinic trajectories and cascade of bifurcations in Lorenz-like systems: birth of strange attractor and 9 homoclinic bifurcations

DOI10.1007/s11071-015-2549-4zbMath1354.37033OpenAlexW2276107102MaRDI QIDQ332823

Gennady Alekseevich Leonov

Publication date: 9 November 2016

Published in: Nonlinear Dynamics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s11071-015-2549-4

zbMATH Keywords

strange attractorChen systemShimizu-Morioka systemLu systemblue sky catastrophecascade of bifurcationsfishing principlehomoclinic trajectoriesLorenz-like systemsaddle valueseparatrix of saddle point

Mathematics Subject Classification ID

Bifurcations of singular points in dynamical systems (37G10) Strange attractors, chaotic dynamics of systems with hyperbolic behavior (37D45) Homoclinic and heteroclinic orbits for dynamical systems (37C29) Homoclinic and heteroclinic solutions to ordinary differential equations (34C37)

Related Items

A family of models with blue sky catastrophes of different classes ⋮ Qualitative behaviors of the continuous-time chaotic dynamical systems describing the interaction of waves in plasma ⋮ Homoclinic Bifurcations and Chaos in the Fishing Principle for the Lorenz-like Systems ⋮ Dynamics of a New 5D Hyperchaotic System of Lorenz Type ⋮ The existence of homoclinic orbits in the Lorenz system via the undetermined coefficient method ⋮ Lyapunov functions in the global analysis of chaotic systems ⋮ On the Homoclinic Orbits of the Lü System ⋮ Unstable periodic orbits analysis in the generalized Lorenz-type system ⋮ Chaotic Dynamics in Generalized Rabinovich System

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• On differences and similarities in the analysis of Lorenz, Chen, and Lu systems
• Formulas for the Lyapunov dimension of attractors of the generalized Lorenz system
• Rössler systems: estimates for the dimension of attractors and homoclinic orbits
• Fishing principle for homoclinic and heteroclinic trajectories
• On bifurcations of the Lorenz attractor in the Shimizu-Morioka model
• A proof that the Lorenz equations have a homoclinic orbit
• On existence of homoclinic paths in the Lorenz system
• A partial differential equation with infinitely many periodic orbits: Chaotic oscillations of a forced beam
• A shooting approach to chaos in the Lorenz equations
• The Tricomi problem for the Shimizu-Morioka dynamical system
• Analytical search for homoclinic bifurcations in the Shimizu-Morioka model
• Criteria for the existence of homoclinic orbits of systems Lu and Chen
• General existence conditions of homoclinic trajectories in dissipative systems. Lorenz, Shimizu-Morioka, Lu and Chen systems
• The Tricomi problem on the existence of homoclinic orbits in dissipative systems
• SHILNIKOV CHAOS IN LORENZ-LIKE SYSTEMS
• On estimates of the bifurcation values of the parameters of a Lorentz system
• A shooting approach to the Lorenz equations
• YET ANOTHER CHAOTIC ATTRACTOR
• A NEW CHAOTIC ATTRACTOR COINED
• Lorenz Equations Part I: Existence and Nonexistence of Homoclinic Orbits