The inverted pendulum: A singularity theory approach

DOI10.1006/jdeq.1998.3623zbMath0936.34029OpenAlexW2098538867MaRDI QIDQ1305487

Igor Hoveijn, Henk W. Broer, Martijn van Noort, Gert Vegter

Publication date: 8 May 2000

Published in: Journal of Differential Equations (Search for Journal in Brave)

Full work available at URL: https://pure.rug.nl/ws/files/3153846/1999JDiffEqBroer.pdf

zbMATH Keywords

Hamiltonian systeminverted pendulumequivariant singularity

Mathematics Subject Classification ID

Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems (37J20) Bifurcation theory for ordinary differential equations (34C23) Normal forms for dynamical systems (37G05) Singular perturbations of ordinary differential equations (34D15) Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol'd diffusion (37J40) Theory of singularities and catastrophe theory (58K99) Stability problems for finite-dimensional Hamiltonian and Lagrangian systems (37J25)

Related Items

Dynamics of a system of coupled inverted pendula with vertical forcing, The Poincaré method: A powerful tool for analyzing synchronization of coupled oscillators, The equations of the approximate theory of the motion of a rigid body with a vibrating suspension point, KAM dynamics and stabilization of a particle sliding over a periodically driven curve, Subharmonic bifurcations and chaotic motions for a class of inverted pendulum system, Asymptotic analysis of resonances in nonlinear vibrations of the 3-dof pendulum, Resonance tongues and instability pockets in the quasi-periodic Hill-Schrödinger equation, The parametrically forced pendulum: A case study in \(1\frac12\) degree of freedom, Excitation-Induced Stability and Phase Transition: A Review, Understanding critical behaviour through visualization: A walk around the pitchfork, Stabilization by vertical vibrations, Resonance tongues in Hill's equations: A geometric approach

Uses Software

• DSTool

Cites Work

• A reversible bifurcation analysis of the inverted pendulum
• Equivariant singularity theory with distinguished parameters: two case studies of resonant Hamiltonian systems
• Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
• The Hamiltonian Hopf bifurcation
• Singularities and groups in bifurcation theory. Volume I
• Singularites \(C^\infty\) en presence de symétrie. En particulier en presence de la symétrie d'un groupe de Lie compact
• A normally elliptic Hamiltonian bifurcation
• Geometrical aspects of stability theory for Hill's equations
• Stability of the Inverted Pendulum—A Topological Explanation
• Resonances in a spring-pendulum: algorithms for equivariant singularity theory
• THE NONLINEAR MATHIEU EQUATION
• Equivalence and Decomposition of Vector Fields About an Elementary Critical Point
• Averaging methods in nonlinear dynamical systems
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item