arXiv subjects

Majid Gazor

Publications and source records attributed to Majid Gazor.

16 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.

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arXiv2023

Symmetry-breaking singular controller design for Bogdanov-Takens bifurcations with an application to Chua system

We provide a complete symmetry-breaking bifurcation control for equivariant smooth differential systems with Bogdanov-Takens singularities. Controller coefficient space is partitioned by critical controller sets into different connected regions. The connected regions provide a classification for all qualitatively different dynamics of the controlled system. Hence, a state feedback controller design with four small controller coefficients is proposed for an efficient and full singular symmetry-breaking control. Our approach works well for nonlinear control systems with both controllable and uncontrollable linearizations. Origin is a primary equilibrium for the uncontrolled system. This gives rise to two secondary local equilibria for the controlled system. These equilibria further experience tertiary fold and hysteresis type bifurcations. The secondary and primary equilibria experience Hopf and Bautin bifurcations leading to the appearance of one limit cycle from primary equilibrium, and either one or two from each secondary equilibria. The collisions of limit cycles with equilibria lead to either a heteroclinic cycle or four different homoclinic cycles. Each pair of limit cycles may respectively merge together and disappear. This is a saddle-node bifurcation of limit cycles. Different combinations of these give rise to a rich list of bifurcation scenarios. Finite determinacy of each of these bifurcations has been thoroughly investigated. This greatly influences their stabilization potential in applications. We consider Chua system with a quadratic state-feedback controller. Controlled Chua system experiences a pitchfork bifurcation, three Hopf bifurcations and two homoclinic bifurcations. There exist two different regions of controller coefficient choices for feedback regularization and two nearby regions for supercritical Hopf stabilization approach.

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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.

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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.

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arXiv2018

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.

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arXiv2018

A user guide for Singularity

This is a user guide for the first version of our developed Maple library, named Singularity. The first version here is designed for the qualitative study of local real zeros of scalar smooth maps. This library will be extended for symbolic bifurcation analysis and control of different singularities including autonomous differential singular systems and local real zeros of multidimensional smooth maps. Many tools and techniques from computational algebraic geometry have been used to develop Singularity. However, we here skip any reference on how this library is developed. This package is useful for both pedagogical and research purposes. Singularity will be updated as our research progresses and will be released for public access once our draft paper [4] is peer-reviewed in a refereed journal.

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arXiv2017

Vector potential normal form classification for completely integrable solenoidal nilpotent singularities

We introduce a sl_2-invariant family of nonlinear vector fields with a non-semisimple triple zero singularity. In this paper we are concerned with characterization and normal form classification of these vector fields. We show that the family constitutes a Lie algebra structure and each vector field from this family is solenoidal, completely integrable and rotational. All such vector fields share a common quadratic invariant. We provide a Poisson structure for the Lie algebra from which the second invariant for each vector field can be readily derived. We show that each vector field from this family can be uniquely characterized by two alternative representations, one uses a vector potential while the other uses two functionally independent Clebsch potentials. Our normal form results are designed to preserve these structures and representations. The results are implemented in Maple in order to compute vector potential and the Clebsch potential normal forms of a given vector field from this family. Some practical normal form coefficient formulas for degrees of up to four are presented.

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arXiv2017

Bifurcation controller designs for the generalized cusp plants of Bogdanov--Takens singularity with an application to ship control

Nonlinear controlled plants with Bogdanov-Takens singularity may experience surprising changes in their number of equilibria, limit cycles and/or their stability types when the controllers slightly vary in the vicinity of critical parameter varieties. Each such a change is called a local bifurcation. We derive novel results with regards to truncated parametric normal form classification of the generalized cusp plants. Then, we suggest effective nonlinear bifurcation control law designs for precisely locating and accurately controlling many different types of bifurcations for two measurable plants from this family. The first is a general quadratic plant with a possible multi-input linear controller while the second is a Z_2-equivariant general plant with possible multi-input linear (Z_2-symmetry preserving) and quadratic (symmetry-breaking) controllers. The bifurcations include from primary to quinary bifurcations of either of the following types: saddle-node, transcritical and pitchfork of equilibria, Z_2-equivariant bifurcations of multiple limit cycles through Hopf, homoclinic, heteroclinic, saddle-node, and saddle-connection, and finally their one-parameter symmetry breaking bifurcations. Using our parametric normal form analysis, we propose a new approach for efficient treatment of tracking and regulating engineering problems with smooth manoeuvering possiblities. Due to the nonlinearity of a ship maneuvering characteristic, there is a need for a controller design in a ship steering system so that the ship follows a desired sea route. The results in bifurcation control analysis are applied to two nonlinear ship course models for such controller designs. Symbolic implementations in Maple and numerical simulations in MATLAB confirm our theoretical results and accurate predictions.

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arXiv2016

Bifurcation control and universal unfolding for Hopf-zero singularities with leading solenoidal terms

In this paper we introduce universal asymptotic unfolding normal forms for nonlinear singular systems. Next, we propose an approach to find the parameters of a parametric singular system that they play the role of universal unfolding parameters. These parameters effectively influence the local dynamics of the system. We propose a systematic approach to locate local bifurcations in terms of these parameters. Here, we apply the proposed approach on Hopf-zero singularities whose the first few low degree terms are incompressible. In this direction, we obtain novel orbital and parametric normal form results for such families by assuming a nonzero quadratic condition. Moreover, we give a truncated universal asymptotic unfolding normal form and prove the finite determinacy of the steady-state bifurcations for two most generic subfamilies of the associated amplitude systems. We analyze the local bifurcations of equilibria, limit cycles and the secondary Hopf bifurcation of invariant tori. The results are successfully implemented and verified using Maple. By employing the proposed approach, we design an effective multiple-parametric quadratic state feedback controller for a singular system on a three dimensional central manifold with two imaginary uncontrollable modes. We illustrate how our program systematically identifies the distinguished (universal unfolding) parameters, derives the estimated transition varieties in terms of these parameters, and locates the local primary and secondary bifurcations of equilibria, limit cycles and invariant tori. This approach is useful in designing efficient nonlinear feedback controllers (single or multiple inputs) for local bifurcation control in engineering problems.

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arXiv2015

Symbolic local bifurcation analysis of scalar smooth maps

The local zero structure of a smooth map may qualitatively change, when the map is subjected to small perturbations. The changes may include births and/or deaths of zeros. The qualitative properties are defined as the invariances of an appropriate equivalence relation. The occurrence of a qualitative change in the zero structures is called a bifurcation and the map is named a singularity. The local bifurcation analysis of singularities has been extensively studied in singularity theory and many powerful algebraic tools have been developed for their study. However, there does not exist any symbolic computer-library for this purpose. We suitably generalize some powerful tools from algebraic geometry for correct implementation of the results from singularity theory. We provide some required criteria along with rigorous proofs for efficient and cognitive computer-implementation. We have accordingly developed a Maple end-user friendly library, named Singularity, for an efficient and complete local bifurcation analysis of real zeros of scalar smooth maps. We have further written a comprehensive user-guide for Singularity. The main features of Singularity are briefly illustrated along with a few examples.

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arXiv2014

Normal Forms of Hopf-zero singularity

The Lie algebra generated by Hopf-zero classical normal forms is decomposed into two versal Lie subalgebras. Some dynamical properties for each subalgebra are described; one is the set of all volume-preserving conservative systems while the other is the maximal Lie algebra of nonconservative systems. This introduces a unique conservative--nonconservative decomposition for the normal form systems. There exists a Lie--subalgebra that is Lie-isomorphic to a large family of vector fields with Bogdanov--Takens singularity. This gives rise to a conclusion that the local dynamics of formal Hopf-zero singularities is well-understood by the study of Bogdanov--Takens singularities. Despite this, the normal form computation of Bogdanov-Takens and Hopf-zero singularities are independent. Thus, by assuming a quadratic non-zero condition, complete results on the simplest Hopf-zero normal forms are obtained in terms of the conservative--nonconservative decomposition. Some practical formulas are derived and the results implemented using Maple. The method has been applied on the Rössler and Kuramoto--Sivashinsky equations to demonstrate the applicability of our results.

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arXiv2013

Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case

We obtain a parametric normal form for any non-degenerate perturbation of the generalized saddle-node case of Bogdanov--Takens singularity. Explicit formulas are derived and greatly simplified for an efficient implementation in any computer algebra system. A Maple program is prepared for an automatic parametric normal form computation. A section is devoted to present some practical formulas which avoid technical details of the paper.

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arXiv2013

Normal forms for Hopf-Zero singularities with nonconservative nonlinear part

In this paper we are concerned with the simplest normal form computation of a family of Hopf-zero vector fields without a first integral. This family of vector fields are the classical normal forms of a larger family of vector fields with Hopf-Zero singularity. Indeed, these are defined such that this family would be a Lie subalgebra for the space of all classical normal form vector fields with Hopf-Zero singularity. The simplest normal forms and simplest orbital normal forms of this family with non-zero quadratic part are computed. We also obtain the simplest parametric normal form of any non-degenerate perturbation of this family within the Lie subalgebra. The symmetry group of the simplest normal forms are also discussed. This is a part of our results in decomposing the normal forms of Hopf-Zero singular systems into systems with a first integral and nonconservative systems.

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arXiv2013

Volume-preserving normal forms of Hopf-zero singularity

A practical method is described for computing the unique generator of the algebra of first integrals associated with a large class of Hopf-zero singularity. The set of all volume-preserving classical normal forms of this singularity is introduced via a Lie algebra description. This is a maximal vector space of classical normal forms with first integral; this is whence our approach works. Systems with a non-zero condition on their quadratic parts are considered. The algebra of all first integrals for any such system has a unique (modulo scalar multiplication) generator. The infinite level volume-preserving parametric normal forms of any non-degenerate perturbation within the Lie algebra of any such system is computed, where it can have rich dynamics. The associated unique generator of the algebra of first integrals are derived. The symmetry group of the infinite level normal forms are also discussed. Some necessary formulas are derived and applied to appropriately modified Rössler and generalized Kuramoto--Sivashinsky equations to demonstrate the applicability of our theoretical results. An approach (introduced by Iooss and Lombardi) is applied to find an optimal truncation for the first level normal forms of these examples with exponentially small remainders. The numerically suggested radius of convergence (for the first integral) associated with a hypernormalization step is discussed for the truncated first level normal forms of the examples. This is achieved by an efficient implementation of the results using Maple.

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arXiv2011

Condensation rank of injective Banach spaces

The condensation rank associates any topological space with a unique ordinal number. In this paper we prove that the condensation rank of any infinite dimensional injective Banach space is equal to or greater than the first uncountable ordinal number.

math.FA↗

arXiv2011

The spectral sequences and parametric normal forms

We generalize recent developments on normal forms and the spectral sequences method to make a foundation for parametric normal forms. We further introduce a new style and costyle to obtain unique parametric normal forms. The results are applied to systems of generalized Hopf singularity with multiple parameters. A different (new) version of this paper has been submitted for a possible publication in a refereed journal.

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