Bifurcation theory has mathematical beginning in the work of Euler ; nevertheless, farther developments were made by Poincare, Andronov, Hopf with the promotions of qualitative theory of differential equations, dynamical system theory, group theory, uniqueness theory etc. A bifurcation in a dynamical system is said to hold occurred when a little smooth alteration in the parametric quantity values ( bifurcation parametric quantities ) of the system causes a ‘qualitative ‘ or topological alteration in its kineticss or behavior. In kernel, a bifurcation occurs when a solutions ‘ behaviour alterations in a extremist manner. The qualitative alterations can sometimes take to complete pandemonium. The methods and consequences of bifurcation theory are cardinal to understanding of nonlinear dynamical systems, and the theory can potentially be applied to many countries.
There are many existent universe examples where, a once stable system begins to hover with larger and larger amplitude when the parametric quantity of the system is changed in one way. Initially stable civil technology constructions, electrical webs, ecological communities could get down to hover and so as the amplitude grows larger, prostration, interrupt down, or remain oscillating when some parametric quantity is varied. The mathematical modeling of such phenomenon leads to differential equations depending on parametric quantities which can so be studied to measure the stableness of the system etc.
Such surveies have led to categorization of Bifurcations into two classs based on where they occur
Global Bifurcation: They frequently occur when big invariant sets of the system collide with each other, or with equilibria of the system. These can non be detected strictly by a stableness analysis of equilibria. A few illustrations of planetary bifurcations include bluish sky calamity,
homoclinic bifurcation, heteroclinic bifurcation and infinite period bifurcation. [ 1 ]
Local Bifurcation: These types of bifurcations are characterized by alterations in local stableness belongingss of equilibria or periodic orbits traversing through critical thresholds. In uninterrupted systems this is tantamount to the existent portion of Eigen value go throughing through nothing.
See a uninterrupted system described by the undermentioned Ode
( 1.1 )
here is the bifurcation parametric quantity.
Categorization of local bifurcations is based on the Eigen values of the Jacobian matrix ( J ) at the fixed point of the system
With the saddle-node, transcritical, and pitchfork bifurcations, the stable fixed point has
P = hint ( J ) & lt ; 0, and q = det ( J ) & gt ; 0. Its characteristic root of a square matrixs are existent and negative ( lie in the left half-plane ) . As I± varies for the Transcritical and Pitchfork bifurcations, one characteristic root of a square matrix becomes positive, turning the fixed point into a Saddle-node.
With Andronov-Hopf bifurcations, the fixed point ‘s characteristic root of a square matrixs are complex conjugates, Aµ1 and Aµ2 with existent portion of the Eigen value being zero. We will be analyzing the sensing of Hopf bifurcation and expression into it in farther item in the undermentioned chapters.
A utile definition in the survey of bifurcation is the Codimension of the bifurcation, which is the figure of parametric quantities to be varied for the bifurcation to happen. Hopf and Saddle node bifurcations are by and large codimension 1, intending they depend merely on one parametric quantity. Most of the others have higher codimensions, but, the normal signifier of the equations after utilizing techniques like Centre manifold decrease become codimension 1.
Brief lineation of anterior work
To observe bifurcations several attacks like Centre manifold decrease and normal signifier theory are available. These involve the undermentioned stairss
Decrease: Designation of the impersonal manner at for a system as defined in ( 1.1 ) so as to curtail the system to the appropriate Centre manifold
Standardization: Application of co-ordinate alterations to the system to do it simple and easier to work with. This cant be generalized and needs to be done for specific instances
Unfolding: Introduction of nonlinear footings which are little and perchance nonlinear to depict the effects of changing off from. These footings are introduced into the normal signifier of the equation.
Shortness: Unfolded system is truncated at some order to help the apprehension of the kineticss of the system. Once the system is understood to an extent the consequence of reconstructing higher order footings is discussed. This consequence is of import in bifurcations like Hopf as the higher order footings are the 1s responsible for qualitative alterations in the system.
The above stairss are hard to implement numerically and analyze systems which are multivariable and non conformable to center multiplex decrease easy. So, an alternate attack was described by Yu.A. Kuznetsov in [ 2 ] . This theory involves description of trial maps which have regular nothing at the uniqueness points like bifurcations. Sometimes a household of trial maps is defined to place a peculiar sort of bifurcation. This was used to develop a GUI called MATCONT which is used in the survey of dynamical systems and detects bifurcations up to codimension 2. The nothing of the trial maps are monitored as the stage secret plan of the system is computed to find the type of bifurcation.
Outline of the study
In this study the ‘objects in inquiry ‘ are dynamical systems in the signifier of differential equations represented in the standard province infinite format. The systems typically studied can be expressed in the signifier of equation ( 1.1 ) after linearization around an equilibrium point.
This study begins with a description of the anterior work done on this subject and the defects of the methodological analysiss soon available. It so follows, to explicate the theory on which the new attack to observe ‘Andronov-Hopf ‘ bifurcation is based. We explore in item Hurwitz stableness trial and its importance in the proposed method.The rudimentss of Semi Dual minimisation and Sum of Squares are studied in deepness before continuing to use the tools available to develop the new algorithm for sensing. We besides analyse and compare assorted strategies of descent algorithms like Newton, Polak-Ribiera, BFGS and gradient.
The study ends with application of the algorithm to two really normally seen illustrations of bifurcation and a man-made which exploits the power of the algorithm to cover with big figure of provinces easy.
Contributions of the undertaking
The parts made to sensing of Hopf bifurcation are three fold
A new algorithm is developed which is simple and does non include calculation of stage portrayals and regular nothing of trial maps
The well established theory- Sum of Squares is exploited to work out the otherwise NP difficult job of non-negativity of univariate multinomials
Small alterations to the algorithm and the construction of the system help us look for stable and unstable Hopf bifurcation in peculiar. This characteristic is non available in any of the commercially available tool chests for survey of uninterrupted systems.
Chapter 2: Background
The recent developments in survey of dynamical systems can be attributed to switching accent from an analytical attack to one with an accent on topology of the orbit construction in stage plane with differential equations. This initial drift triggered the existent detonation of involvement for dynamical systems and allow to valuable parts by Smale and Arnold in the sixtiess.
To acknowledge the particular types of kineticss which has pervert construction the planetary attack of the 1960ss by the Smale school was of import. The planetary attack further led to categorization of a generic set of systems, which now could be described rather merely.
This in bend gave a new drift to the work of Andronov et Al. [ 3 ] , Neimark, Sacker, and Hopf on households of dynamical systems. The anticipation of planetary phenomenon utilizing local computations was now possible. For case, Hopf bifurcations predict the growing of a bound rhythm or invariant circle around an equilibrium point which undergoes a alteration of stableness. There are nondegeneracy conditions associated with the creative activity of this circle, but they are all calculated at the fxed point.
It is about impossible to hold an thought beforehand which of the parametric quantities is critical to blossoming the debauched behavior of the shaping equations. Of class, some parametric quantities may hold no structural consequence on the system whatsoever. Often a dynamical system patterning a real-life phenomenon comes prepackaged with excessively many parametric quantities to manage. It is really dashing to seek in large-dimensional parametric quantity infinites! The growing and categorization of bifurcation has been highly valuable in leting modelers to concentrate their hunt for cardinal dynamical behavior in the parametric quantity infinite.
Deriving penetration into the bifurcation behavior of a system and obtaining a complete set of topological types of stage portrayal that occur by changing parametric quantities is an vastly utile usher to the behavior of the system. This cognition can be used to command the system utilizing the emerging bifurcation control techniques.
Most of the work that has been done and mentioned above has been good documented in [ 2 ] assisting us analyze the background in item.
This chapter begins with the basic Hopf theorem in the clip sphere followed by an application which was based on it. It so explores briefly the Graphical Hopf Theorem without traveling into strict mathematical cogent evidence as we do n’t necessitate it further.
2.2 Hopf theorem
Let us get down our survey of anterior work done with the basic Hopf theorem. The theorem is as follows:
Theorem 2.1 ( Hopf [ 4 ] )
See the system
where and ; say that for little the matrix has a brace of complex conjugate characteristic root of a square matrixs ( the derived function of the existent portion with regard to the parametric quantity Aµ is positive ) and the other n-2 characteristic root of a square matrixs have negative existent portion ; so
there exist a and a map such that for the system
has a periodic solution with period T ( besides, , and the amplitude of this periodic solution ( the approximate distance of the corresponding periodic orbit from the beginning ) is relative to ;
the beginning of the infinite has a vicinity that does non incorporate any periodic orbit of ( 2.1 ) but those of the household ;
if the beginning x=0 is a 3-asymptotically stalls ( resp. 3-unstable ) 3-unstable equilibrium of the system ( resp. for and the periodic solution is asymptotically orbitally stable ( resp.unstable )
The above theorem with the statements ( I ) , ( two ) and ( three ) province the being, the singularity and the stableness of the bifurcating periodic solution, severally. The singularity is to be understood as it is expressed in ( two ) and it is to be observed that it does non intend that to each sufficiently little there belongs precisely one periodic orbit. The point is that the map may presume the same value at different values of arbitrary near ( cf.Negrini-Salvador [ 1979 ] ) ensuing in several periodic orbits matching to the same system ( to the same ) but to different ‘s, or even it may go on at
INSERT FIGURE 2.2.1 ( scan and paste )
There are two typical conditions that arise if the status in ( three ) is satisfied. These are shown in the Figure 2.2.1. The first instance when x=0 is 3-asymptotically stable is called a supercritical bifurcation since so the periodic orbits appear for, i.e. above the critical value of the bifurcation parametric quantity. The 2nd instance when x=0 is 3-unstable is called subcritical bifurcation: in this instance the periodic orbits appear below the critical value, i.e. for negative values of
Sometimes, the several footings “ soft and difficult bifurcations ” are used: “ soft ” because so the system begins to hover stably with little amplitudes ; “ difficult ” because so the behavior of the system is unforeseeable, normally is starts to hover with Wilder and Wilder amplitudes taking frequently to breakdown/collapse/chaos.
Theorem 1 signifiers the footing of the Hopf bifurcation sensing procedure with its tremendously of import being standards.
2.3 Detection methodological analysiss
One of the most popular techniques that have been exploited to analyze dynamical system is that of ‘test maps ‘ . We will try to understand how this is used in the context of a GUI-MATCONT [ 5 ] which was developed to analyze dynamical systems. The undermentioned account is based on the work done in [ 2 ] and other beginnings
2.3.1 Calculation of solution curve
Let us see a smooth map
( 2.2 )
A solution curve of the equation F ( x ) =0 is computed with the aid of a technique called ‘Numerical Continuation ‘ .This technique obtains a series of points which approximate a given subdivision. Most continuance algorithms implement a predictor-corrector method. The thought behind this method is to bring forth a sequence of points xi, i=1,2, … n along the curve, fulfilling a chosen tolerance standard: ||F ( xi ) || a‰¤ E› for some E› & gt ; 0 and an extra truth status ||dxi|| a‰¤ E› where E› & gt ; 0 and dxi is the last Newton rectification.
The coevals of new points consists of two of import stairss
Prediction of new points is made with normally used tangent anticipation technique
Correction of the generated points
We assume that is near to the curve. To happen the point xi+1 on the curve we use a Newton-like process. Since the standard Newton loops can merely be applied to systems with the same figure of equations as terra incognitas and excess scalar status is appended. This status is based on ‘Moore-Penrose ‘ discharge length anticipation.
Therefore, we now have a methodological analysis available for calculating the solution curve. It remains now to discourse the trial maps which need to be monitored as the curve is being computed to happen out the bifurcations in the stage secret plan. The undermentioned bomber subdivisions give inside informations of the trial maps and how they are combined to organize a uniqueness matrix to supervise different sorts of bifurcations.
2.3.2 Test maps
The chief thought is to observe uniquenesss by specifying smooth scalar maps which have regular nothing at the uniqueness points. These maps are called trial maps. Suppose we have a uniqueness S which is noticeable by a trial map
Let us farther presume that we find two points on the curve
The uniqueness S will be detected merely if
Having found two points we now locate a point vanishes. An obvious manner of making this is work outing the undermentioned equations utilizing Newton ‘s loop starting at the point
However, it is necessary to calculate the derived functions of F ( x ) to utilize this method which is non ever to calculate. So, a unidimensional secant method to turn up degree Fahrenheit ( x ) =0 along the curve was implemented by the coders as a default one.
The above method is a general manner to observe and turn up uniquenesss depending on one trial map the cogent evidence for which is given in greater item in [ 2 ] . However, it may go on that it is non possible to stand for a uniqueness with merely one trial map.
Let us now suppose there is a uniqueness S which depends on nt trial maps which all alteration mark at a sequence for some back-to-back points xi and xi+1:
A one dimensional secant hunt gives all the nothing of the trial maps which in the ideal instance should co-occur. Since the continuance is numeral and non exact we tend to happen nothing which are clustered around a Centre point
A bunch is so detected by happening the mean of all the nothing of the trial maps. Conditionss for observing the bunch and the expression for obtaining the Centre are as follows:
This helps us specify x* as a mean of all the nothing of the trial maps
2.3.3 Singularity matrix
Until now we have dealt with uniquenesss which vanish at one trial maps. When a big figure of uniquenesss are monitored we need a amalgamate representation uniting the trial maps and where they vanish. The representation that follows has been adopted from [ 6 ]
Say we have two uniquenesss S1 and S2, depending severally on trial maps and.Further, assume that vanishes at both S1 and S2, while vanishes at merely S2. Therefore we may necessitate to stand for uniquenesss with non disappearing trial maps as.
To stand for all uniquenesss we introduce a uniqueness matrix ( as in [ 6 ] ) . This matrix is a compact manner to depict the relation between the uniquenesss and all trial maps. Suppose we are interested in uniquenesss and trial maps which are needed to observe and turn up the uniquenesss. Let S be a matrix of dimension
There is besides a possibility of the default location algorithm running into jobs for which there is an option provided for the user to stipulate their ain trial map which is monitored.
2.4 Graphic Hopf Theorem-a position ( GHT )
Hopf [ 4 ] showed that oscillations near an equilibrium point can be understood by looking at the characteristic root of a square matrixs of the linearized equations for disturbances from equilibrium, and at certain important derived functions of the equations. The graphical standard checks the Hopf conditions for the being of stable or unstable periodic oscillations. Since it is evocative of the generalised Nyquist standard for additive systems, graphical process can be interpreted as the frequence sphere version of the Hopf bifurcation theorem.
Without traveling into strict mathematical cogent evidence, allow us seek to understand the GHT with its similarity to the clip domain version. [ 7 ]
See a general n-dimensional linear system with nonlinear feedback-u as follows:
with an initial status x ( 0 ) =0. In ( 2.13 ) is a ( variable ) existent parametric quantity, A is a system matrix, B and C matrices severally, and nonlinear map which is used as province feedback. After a few stairss which involve Laplace transform of the end product vector Y and uses we can deduce the equation
where represents the Laplace transform of the signal Y and
It so follows from equation ( 2.14 ) that we may merely cover with ( contained in etc. ) in the frequence sphere, without straight sing the states/ of the system.
The frequence sphere version of Hopf bifurcation theorem will hold a lower dimension as m & lt ; n in feedback systems. This is one of the advantages over the time-domain parallel.
If we proceed to linearise the system by measuring its Jacobian – about the equilibrium end product we obtain an equivalent of the clip domain version of the theorem as follows:
Lemma 1.If an characteristic root of a square matrix of the corresponding Jacobian of the nonlinear system ( 2.13 ) , in the clip sphere, assumes a strictly fanciful valueat a peculiar value, so the matching characteristic root of a square matrix of the changeless matrix in the frequence sphere must presume the value at
The above stated Lemma along with a generalised signifier of Nyquist standards of stableness is the footing of the GHT which was used by Allwright-Mees-Chua ( Allwright [ 1977 ] , Mees & A ; Chua [ 1979 ] , and Mees [ 1981 ] ) to develop the graphical version. The theorem as such is non being stated here as we will non be utilizing it further.
The equality of both theorems with the pros and cons of both has been discussed extensively in [ 7 ] .
2.5 Central Manifold Theorem
We are traveling to explicate without cogent evidence the chief theorem that allows us to cut down the dimension of a given system near a local bifurcation. Let us get down with the critical instance ; we assume that the system values are fixed at their bifurcation values, which are those for which there is a nonhyperbolic equilibrium. The theorem which is traveling to be stated is widely used to curtail the order of the system to the Centre manifold and therefore cut downing the complexness of the system. It makes the calculation easier when trial maps are used.
See a uninterrupted clip dynamical system [ 2 ]
where degree Fahrenheit is sufficiently smooth, f ( 0 ) =0. Let the characteristic root of a square matrixs of the Jacobian matrix A evaluated at the equilibrium point. Suppose there are characteristic root of a square matrixs with zero existent portion doing the equilibrium nonhyperbolic. Assume that there are characteristic root of a square matrixs with, characteristic root of a square matrixs with, and characteristic root of a square matrixs with. Let denote the additive eigenspace of A matching to the brotherhood of the characteristic root of a square matrixs on the fanciful axis. The characteristic root of a square matrixs with are frequently called critical, as is the eigenspace. Let correspond to the flow associate with ( 2.15 ) . Under these premises the undermentioned theorem holds. INSERT PIC
There is a locally defined smooth -dimensional invariant manifold of ( 2.15 ) that is tangent to at x=0.
Furthermore, there is a neighbourhood U of, such that if for all so, for
The manifold is called the Centre manifold.
There are certain tax write-offs that can be made from the Theorem 2.2 which are utile. We can reason that the Centre manifold is pulling from the 2nd statement in the theorem. A really critical fact is the non-uniqueness of the Centre manifold. It is supposed to hold the same finite smoothness as
degree Fahrenheit in some neighbourhood U of x0. Besides, this theorem has an tantamount notation and definition in the distinct clip instance.
The above stated can therefore be used to cut down the system to the Centre manifold and survey it at that place. There are other theorems by ( Shoshitaishvili [ 1972 ] ) which prove the topological equality of the system when it is restricted to the Centre manifold.
In parametric quantity dependant system ( 2.1 ) like the 1s we deal with the centre manifolds are of equal importance. The manifold is tangent at the beginning to the ( generalized ) eigenspace of J matching to characteristic root of a square matrixs with zero existent portion. Since, we have invariant hyperplanes which foliate the manifold.
where are invariant hyperplanes.
INSERT PIC.. ! ! ! ! !
2.6 Defects of anterior work
There are a few defects in the trial map attack as described above and in [ 2 ] , [ 5 ] and [ 6 ] .Firstly, a batch of calculation is involved because the solution curve is calculated to observe the bifurcation. It might be easier in most instances to find the happening of bifurcation without needed the information of how the solution evolves. The characterization/defining characteristics of a peculiar type of bifurcation is non exploited in this strategy.
Second, the cognition of ‘when and where ‘ the bifurcation occurs is of more as a batch of control strategies can be developed around this cognition. It gives the control system designer adequate information to chalk out the control strategy.
Third, it does non let us to happen out the nature of bifurcation i.e. whether it is stable or unstable. This information is really of import to analyze the system further and do alterations if we require stableness
Fourthly, we are non cognizant of precisely at what value of parametric quantity the bifurcation occurs. We get information about the type of bifurcation and at what point on the solution curve it occurs, it is non straightforward to happen out at what value it occurs.
Furthermore, from the GHT point of position the analysis of the system is non straightforward and non much theory has been developed to help us. In peculiar, many utile methods and important consequences have been developed in the clip sphere puting for analysing dynamical behavior of nonlinear ODEs arising from remarkable ( particularly multiple ) bifurcations every bit good as period duplicating sequences, yet their corresponding frequence domain graphical version has non been completed. Besides, even though GHT reduces the order of the system the characteristic equation of the lower-order multinomial is instead complicated. It appears to be an algebraic map, which has the frequence variable I‰ as a parametric quantity of the characteristic venue adding to the complications. With all these disadvantages in head it is rather apparent why this attack to the job was non considered.
2.7 Concluding comments
The theorem 2.1 stated at the beginning of the chapter is of extreme importance in this study. It non merely discusses the fortunes conducive for the happening of Hopf bifurcation but besides provides conditions for stableness and singularity.
The defects stated in subdivision 2.4 indicate the demand for a simpler more efficient sensing methodological analysis. This study fulfils that demand by working the characteristics of Hopf bifurcation and utilizing a simple technique to observe their happening. Hence, most of the drawbacks are overcome in the new algorithm proposed.A similar process which takes advantage of features of the bifurcation can be employed for others besides.
Chapter 3: New Approach to the job
The old chapter discussed in some item the anterior work done on the job of sensing of bifurcation and the defects of it. We begin this chapter by reexamining the basic Hopf theorem and catching a little glance of the new algorithm devised. This chapter so explores the Hurtwitz multinomial and SOS in item before discoursing the imposter codification for the algorithm.
3.2 Footing of the new attack
Most of the old work which was concerned with bialternate merchandises of matrices, Nyquist theorems and trial maps did non work the simple stableness issue that arises when there is a bifurcation in a dynamical system. A Hurwitz multinomial is obtained as the determiner of the Hurwitz matrix. This matrix is formed with the coefficients of the characteristic multinomial of the system matrix of a system as defined in ( 2.1 ) .Change in stability/the motion of Eigen values can be characterized by the alteration in mark of the Hurwitz multinomial from positive to negative or frailty versa.
In the instance of Hopf bifurcation we observe the Hurwitz determiner going nothing at the critical value of the bifurcation parametric quantity.
The mark alteration in the determiner can easy be monitored by taking advantage of the relation between positiveness and look of a multinomial as a amount of squares multinomial. The really celebrated Hilbert ‘s 17th job concerns itself with the question- ” Given a multivariate multinomial that takes merely non-negative values over the reals, can it be represented as a amount of squares of rational maps? ”
The inquiry was answered with an affirmatory, shown by Emil Artin in 1927. An algorithm for happening the rational map was subsequently formulated and popularised. Over the old ages, nevertheless, a few counter illustrations were found to this preparation. Explicit sufficient conditions for a multinomial to be a amount of squares of other multinomials were found by Wormann and Powers. We explore this relation in the new attack to the job
The positiveness of the Hurwitz determiner over the reals is a really good manner of guaranting the non happening of a bifurcation. In the instance of univariate multinomials like the 1s we are concerned with, the relation between positiveness and amount of squares is steadfastly established. Therefore, we can be certain that if there exists is a amount of square decomposition for the Hurwitz determiner so we remove the possibility of a bifurcation go oning. This decision was used to develop the algorithm besides taking into history that the bifurcation happens from an equilibrium point of the dynamical system
3.3 Characteristic multinomial of a matrix
3.4 Hurwitz-stability trial
Hurwitz has related the stableness of an nth order multinomial
to a set of determiners The determiners come from the following set of n Hurwitz matrices:
This form continues until an nA-n matrix is obtained. For n even, this last matrix Hn has the signifier
for n odd, it has the signifier
As an illustration, if the multinomial ( 3.1 ) is of order 4, the largest Hurwitz matrix is
while for n=5, the largest Hurwitz matrix is
The relation of these matrices to stableness is given in the undermentioned theorem. There will be no cogent evidence shall be provided for this well-known consequence of Hurwitz.
An nth order multinomial ( 3.1 ) is stable if and merely if
Note that, for both n even and n odd, it follows from spread outing by its last column that
Therefore, the stableness status 3.5 is tantamount to for, and a0 & gt ; 0.
From the above theorem we can reason that the lone determiner we need to look into for positiveness is. Given the characteristic multinomial of the system matrix for ( 2.1 ) we can build the Hurwitz matrix and cipher the deciding quite easy in any programming linguistic communication. Matlab was used to make the above mentioned operation.
The following subdivision describes the implicit in rules of proving the positiveness of the Hurwitz ‘s determiner utilizing tools and techniques which are good established in theory and in practise but have non been applied as yet to bifurcation analysis.
3.5 Nonnegativity and amounts of squares ( SOS )
A univariate multinomial P ( ten ) ( in one variable or in one bifurcation parametric quantity ) is said to be a amount of squares ( SOS for brevity ) if there exist such that [ 8 ]
If a multinomial P ( x ) is a amount of squares, so it evidently satisfies for all
Therefore, this status is a sufficient 1 for planetary nonnegativity. It can besides be shown that the converse is true in the univariate instance entirely.
A univariate multinomial is nonnegative if and merely if it is a amount of squares.
Theorem 3.2 once more is of extreme importance to us because it deals with nonnegativity of a multinomial. The nonnegativity of the can be comparatively easy determined by utilizing this technique as it is manipulable or determinable ; otherwise this job is a NP hard. NP difficult jobs are difficult to work out but, it is easy to demo if a plausible solution is a true solution or non in multinomial clip. The categorization of this job as NP hard is easy to see ; the whole of existent infinite demands to be spanned if we can non show it as amounts of squares. A Turing Machine may ne’er halt if the job of look intoing nonnegativity is fed to it. Besides, note that the theorem is peculiarly applicable merely for a univariate instance. In the instance when multinomials under consideration have more than one variable the relaxations which we talk about subsequently on may non be justified and we may non obtain a direct relation between nonnegativity of the multinomial and showing it as amounts of squares.
Over the old ages a few counter illustrations were found where the multinomial is nonnegative but, it could non be expressed as amounts of squares. In theory, we do cognize that a multinomial of any grade and any figure of variables can be expressed as SOS of rational maps. The widely accepted cogent evidence of this Hilbert ‘s seventeenth job by Artin was given rather a long clip ago. However, it is hard to plan the process of happening a SOS look in the generic instance due to the computational restrictions and relaxations. These considerations and exclusions will be discussed subsequently on in this study.