arXiv subjects

Ahmad Shoghi

Publications and source records attributed to Ahmad Shoghi.

4 recordsLinked to original sources

arXiv2023

Further elements on hypernormal forms of non-resonant double Hopf singularities

In this paper, we deal with hypernormal forms of non-resonant double Hopf singularities. We investigate the infinite level normal form classification of such singularities with nonzero radial cubic part. We provide a normal form decomposition of normal form vector fields in terms of planar-rotating and planar-radial vector fields. These facilitate the pattern recognition and analysis of the corresponding generalized homological maps. This paper is the first instance of the normal form classification for generic non-resonant double Hopf singularities without structural symmetry.

math.CA↗

arXiv2020

Musical tone coloring via bifurcation control of Eulerian n-tuple Hopf singularities

An intrinsic essence of sounds in music is the evolution of their qualitative types while in mathematics we interpret each qualitative change by a bifurcation. Hopf bifurcation is an important venue to generate a signal with an arbitrary frequency. Hence, the investigations of musical sounds via bifurcation control theory are long-overdue and natural contributions. In this paper, we address the tone coloring of sounds by dynamical modeling of spectral and temporal envelopes. Multiple number of leading harmonic partials of a note (modulo a hearing sound velocity threshold) are attributed into an Eulerian differential system with n-tuple Hopf singularity. The qualitative evolution of the temporal envelop is then simulated over a set of consecutive time-intervals via bifurcation control of the differential system. For an instance, our proposed approach is applied on audio C#4 files obtained from piano and violin. Fourier analysis is used to generate the amplitude spectral vectors. Then, we associate each amplitude spectral vector with an Eulerian flow-invariant leaf. Bifurcation control suffices to accurately construct the desired spectral and amplitude envelopes of musical notes. These correspond with a rich bifurcation scenarios involving Clifford toral manifolds for the Eulerian differential system. In order to reduce the technicalities, we employ several reduction techniques and use one bifurcation parameter. We show how different ordered sets of elementary bifurcations such as pitchfork and (double) saddle-node bifurcations are associated with the qualitative temporal envelop changes of a C]4 played by either a piano or a violin. A complete hysteresis type cycle is observed within the temporal envelop bifurcations of the C#4 played by violin.

math.DS↗

arXiv2020

Toral CW complexes and bifurcation control in Eulerian flows with multiple Hopf singularities

We are concerned with bifurcation analysis and control of nonlinear Eulerian flows with non-resonant n-tuple Hopf singularity. The analysis is involved with CW complex bifurcations of flow-invariant Clifford hypertori, where we refer to these toral manifolds by toral CW complexes. We observe from primary to tertiary flow-invariant toral CW complex bifurcations for one-parametric systems associated with two most generic cases. In a particular case, a tertiary toral CW complex bifurcates from and resides outside a secondary toral CW complex. When the parameter varies, the secondary internal toral CW complex collapses with the origin. However, the tertiary external toral CW complex continues to live even after the secondary internal toral manifold disappears. Our analysis starts with a flow-invariant primary cell-decomposition of the state space. Each open cell admits a secondary cell-decomposition via a smooth flow-invariant foliation. Each leaf of the foliations is a minimal flow-invariant realization of the state space configuration for all Eulerian flows with n-tuple Hopf singularities. Permissible leaf-vector field preserving transformations are introduced via a Lie algebra structure for nonlinear vector fields on the leaf-manifold. Complete parametric leaf-normal form classification is provided for singular leaf-flows. Leaf-bifurcation analysis of leaf-normal forms are performed for three most leaf-generic cases associated with one to three unfolding bifurcation-parameters. Leaf-bifurcation varieties are derived. Leaf-bifurcations provides a venue for cell-bifurcation control of invariant toral CW complexes. The results are implemented and verified using Maple for practical bifurcation control of such parametric nonlinear oscillators.

math.DS↗

arXiv2019

Parametric normal form classification for Eulerian and rotational non-resonant double Hopf singularities

In this paper we provide novel results on the infinite level normal form and orbital normal form classifications of nonlinear Eulerian and rotational vector fields with two pairs of non-resonant imaginary modes. We use the method of multiple Lie brackets and its extension along with time rescaling for orbital normal form classification. Furthermore, we apply two reduction techniques. The first is to use the radical Lie ideal of rotational vector fields and its corresponding quotient Lie algebra. The second technique is to employ a Schur complement block matrix type in Gaussian elimination and analysis of block matrices. The infinite level parametric normal form classification are also presented. The latter is also viewed as a normal form result for multiple-input controlled systems with non-resonant double Hopf singularity. We also discuss nonlinear symmetry transformations associated with the nonlinear symmetry group of the simplest normal forms. Symbolic normal form transformation generators are derived for computer algebra implementation. Further, the results are efficiently implemented and verified using Maple for all three types of normal form computations up to arbitrary degree, where they can also include both small bifurcation parameters and arbitrary symbolic constant coefficients.

math.OC↗