The Hopf Bifurcation and Its Applications

By Jerrold E. Marsden, M. McCracken, Springer New, York Berlin. 1976

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1. Junjie Wei
2. Junping Shi
3. Shigui Ruan
4. Wenwu Yu
5. Zhihua Liu
6. Weihua Jiang
7. Jinde Cao
8. Pierre Magal

Citations in the corpus

• Perturbations and Hopf bifurcation of the planar discontinuous dynamical system (2004, unknown): “The present paper can be considered as an attempt to apply ideas of the perturbations theory, which was founded by Poincaré and Lyapunov [28,33], and methods of the bifurcation theory[4,8,15,18,21,29,30,33] to the object which combines features of vector fields and maps.”
• Formulation of the normal form of Turing-Hopf bifurcation in partial functional differential equations (2020, background): “Often a center manifold reduction reduces a higher dimensional problem to a lower dimensional one, and the normal form can be computed on the center manifold [9,26,40, 57].”
• Computing and Stability in Cortical Networks (2004, methodology): “To simplify our analysis, we adopt the the effective response function approach ofMascaro and Amit (1999), which can be implemented by taking the limitof fast inhibition, ¿I ! 0.”
• Cones of rank 2 and the Poincaré–Bendixson property for a new class of monotone systems (2009, background): “The proofs can be found in [3] and [16].”
• 2:1 Resonance in the delayed nonlinear Mathieu equation (2007, background): “This phenomenon is well-known [13], and a condition has been given that guarantees that no such degeneracy will occur, namely that the equilibrium point should be a “vague attractor” (or less generally, asymptotically stable) when μ = 0.”
• Methods to reduce non-linear mechanical systems for instability computation (2004, methodology): “One of the most useful non-linear methods to reduce systems at near-equilibrium point is the centre manifold approach ([135], [70], [99], [82], [84], [62], [44], [28] [116], [167], [170] and [172]).”
• Coupling functions:universal insights into dynamical interaction mechanisms (2017, methodology): “…Pikovsky et al., 2001), oscillation and amplitude death (Koseska et al., 2013a; Saxena et al., 2012), birth of oscillations (Pogromsky et al., 1999; Smale, 1976), breathers (MacKay and Aubry, 1994), coexisting phases (Keller et al., 1992), fractal dimensions (Aguirre et al., 2009), network…”
• Dynamical behavior of a delay virus dynamics model with CTL immune response (2010, methodology): “To apply the Hopf bifurcation theorem as stated in [15], we state the following lemma.”
• Hopf bifurcation and stability of periodic solutions for van der Pol equation with time delay (2005, background): “This occurs when a eigenvalue crosses the imaginary axis from left to right as a real parameter in the equation passes through a critical value [15,21].”