Catastrophe Theory and Its Applications

Poston, Timothy; Stewart, Ian; Plaut, Raymond H.

doi:10.1115/1.3424591

Cited by 655 publications

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“…This also follows from computations as in Arnold [3,6] in this symplectic setting, e.g. compare Broer [10], as well as from the codimension computations in Singularity Theory, carried out on the level of 2-jets, again see [9,22,28,37,45]. Note, that by 'normal form' we just mean a preferred, simple member of the corresponding equivalence class.…”

Section: The Linear Classificationmentioning

confidence: 99%

“…This question is addressed by Singularity Theory, for details we again refer to [9,22,28,37,45]. The codimension 0 cases i and ii can be handled by the Morse Lemma, which implies that the normal forms given above, also hold with the higher order terms added.…”

Section: The Fold and The Elliptic And Hyperbolic Umbilic Catastrophesmentioning

confidence: 99%

Section: H 8h =Tf' Y-mentioning

confidence: 99%

“…The characteristic polynomial of a matrix L ~ sl(2, ~) is given by 22+ det(L), which directly leads to the following partition in equivalence classes; for similar classifications e.g. compare Gibson [22] and Poston and Stewart [37]. iv.…”

Section: The Linear Classificationmentioning

confidence: 99%

“…In the former of these cases the corresponding Hamiltonian H, at the origin, vanishes to third order. We claim that the Singularity Theory for the planar Hamiltonian functions provides a good framework for this problem, again compare [9,22,28,37,45].In fact, we start considering a Coo transformation @: R2~ ~2 such that for two Hamiltonians H and K one has H = K o aS. Such a map ~ is called a right equivalence between the functions H and K. (O, XH, ").…”

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A normally elliptic Hamiltonian bifurcation

Broer

Chow

Kim

et al. 1993

Z. angew. Math. Phys.

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The unfolding theory, to be developed below, thenThe conjugate pair of imaginary eigenvalues gives rise to a formal rotational symmetry in all unfoldings, i.e. a rotational symmetry in their Taylor series. Here the series is considered in dependence on both the phase space variables and the parameters. This is an application of Normal Form Theory, where the terms of the formal power series are changed by canonical coordinate transformations in an inductive process. The symmetry, thus obtained, enables a formal reduction to 1 degree of freedom, around an equilibrium point with a double eigenvalue 0. For similar approaches see some of the above references, also compare e.g. Arnold [6], Takens [43,44], Broer [10,11], Golubitsky and Stewart [23] and Broer and Vegter [16]. This Normal Form Theory will be presented in Section 3.In Section 2, we begin studying the 1 degree of freedom 'backbone'-problem in its own right. Since in the plane the integral curves of the systems are the level sets of the Hamiltonian functions, here the problem reduces to Singularity Theory, e.g. In Section 4 the connection is made between the planar reduction of the symmetric system and the 'backbone'-system of Section 2. It turns out that the formal integral, obtained by normalization, is a distinguished parameter in the sense of Golubitsky and Schaeffer [51] and Schecter [41], also compare Wassermann [52]. Now Singularity Theory yields new normal forms, that are polynomial of degree 3, at least in the phase-space variables. We note that here the normalizing transformations no longer need to be canonical. Technical details from Singularity Theory have been collected in an appendix (Section 7). After this we suspend, or dereduce, to the original 4-dimensional setting, so obtaining an integrable, i.e. rotationally symmetric, approximation of the original unfolding.Thus we obtain a perturbation problem, similar to e.g. Broer [10,12] or Braaksma and Broer [8]. The perturbation term is of arbitrarily high order, both in the phase space variables and the parameters. This problem is briefly addressed in Section 5.Remarks. (i) Although in the original family only generic restrictions are imposed on the lower order terms, by Normal Form Theory and Singularity Theory, the corresponding unfolding is reduced to an arbitrarily flat perturbation of a normal form, completely determined by a 1 degree of freedom system, which is polynomial of degree 3. This polynomial character Vol. 44, 1993 A normally elliptic Hamiltonian bifurcation 391 may explain why certain 'integrable' characteristics in the unfoldings are so persistent. For a related comment we refer to Verhulst [49].(ii) Our approach differs from e.g. Van der Meer [46], also compare Duistermaat [19]. In [46,19] the dynamics is reduced by the energy-momentum map as well as the Moser-Weinstein method. This gives information related to specific periodic solutions, that is also important for the dynamics as a whole. Instead, we just factor out a formal rotational symmetry, and so seem to get a more ...

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Section: The Linear Classificationmentioning

confidence: 99%

Section: The Fold and The Elliptic And Hyperbolic Umbilic Catastrophesmentioning

confidence: 99%

Section: H 8h =Tf' Y-mentioning

confidence: 99%

Section: The Linear Classificationmentioning

confidence: 99%

mentioning

confidence: 99%

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A normally elliptic Hamiltonian bifurcation

Broer

Chow

Kim

et al. 1993

Z. angew. Math. Phys.

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Moving From Static to Dynamic Models of the Onset of Mental Disorder

et al. 2017

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Psychiatric research may benefit from approaching psychopathology as a system rather than as a category, identifying dynamics of system change (eg, abrupt vs gradual psychosis onset), and determining the factors to which these systems are most sensitive (eg, interpersonal dynamics and neurochemical change) and the individual variability in system architecture and change. These goals can be advanced by testing hypotheses that emerge from cross-disciplinary models of complex systems. Future studies require repeated longitudinal assessment of relevant variables through either (or a combination of) micro-level (momentary and day-to-day) and macro-level (month and year) assessments. Ecological momentary assessment is a data collection technique appropriate for micro-level assessment. Relevant statistical approaches are joint modeling and time series analysis, including metric-based and model-based methods that draw on the mathematical principles of dynamical systems. This next generation of prediction studies may more accurately model the dynamic nature of psychopathology and system change as well as have treatment implications, such as introducing a means of identifying critical periods of risk for mental state deterioration.

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