On Algebraic Approach in Quadratic Systems

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Volume 2011, Article ID 230939,12pages doi:10.1155/2011/230939

Review Article

On Algebraic Approach in Quadratic Systems

Matej Mencinger

^{1, 2}

1Department of Basic Science, Faculty of Civil Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia

2Department of Mathematics, Institute of Mathematics, Physics and Mechanics Ljubljana, 1000 Ljubljana, Jadranska 19, Slovenia

Correspondence should be addressed to Matej Mencinger,[email protected] Received 14 December 2010; Accepted 9 February 2011

Academic Editor: Ivan Chajda

Copyrightq2011 Matej Mencinger. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

When considering friction or resistance, many physical processes are mathematically simulated by quadratic systems of ODEs or discrete quadratic dynamical systems. Probably the most important problem when such systems are applied in engineering is the stability of critical points and nonchaotic dynamics. In this paper we consider homogeneous quadratic systems via the so- called Markus approach. We use the one-to-one correspondence between homogeneous quadratic dynamical systems and algebra which was originally introduced by Markus in1960. We resume some connections between the dynamics of the quadratic systems andalgebraicproperties of the corresponding algebras. We consider some general connections and the influence of power- associativity in the corresponding quadratic system.

1. Introduction

The stability of hyperbolic critical points in nonlinear systems of ODEs is well-known.

It is described by the stable manifold theorem and Hartman’s theorem. The critical or equilibrium or stationary or fixed point of x fx or xk1 fxk is defined to be the solution of the following algebraic system of equations,fx0 0 or fx0 x0, respectively. For the systems of ODEs,xfx, the critical pointx0is said to be hyperbolic if no eigenvalue of the corresponding Jacobian matrix,Jfx0, of thenonlinear vectorfunction fhas it is eigenvalue equal to zeroi.e., Reλ_i/0. In case of discrete system,x_k1 fx_k, the critical pointx0 is said to be hyperbolic if no eigenvalue of the Jacobian matrix has it is eigenvalue equal to 1i.e.,|λi|/1. Roughly speaking, if for a continuous system Reλi<0 for everyλ_i, the corresponding critical point is stableit is unstable, if Reλ_i>0 for someλ_i. Similar, if for discrete systems|λi|<1 for everyλi, the corresponding critical point is stableit is unstable, if|λi|>1 for someλi. Note that just one eigenvalue of the corresponding linear approximation of x fxor x_k1 fx_kfor which Reλ_i 0 or|λ_i| 1, respectively,

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implies that the stability must be investigated separately in each particular casebecause of the significance of the higher order terms. Such articles where for the non-hyperbolic critical points the classes of stable and unstable systems are considered are published constantly.

The authors consider the influence of at least quadratic terms added to the linear ones.The most recent article on quadratic systems might be1. For homogeneous quadratic systems the origin is an example of the so-called totally degenerated i.e., non-hyperbolic critical point.

In this paper the algebraic approach to autonomous homogeneous quadratic continuous systems of the formxQxand autonomous homogeneous quadratic discrete dynamical systems of the form xk1 Qxk where Q : IRⁿ → IRⁿ is homogeneous of degree two in each component:Qax a²Qxfor eachrealais considered, as suggested by Markus in2. Markus idea was to define a unique algebra multiplication via the following bilinear formBx, y x∗y:

x∗y: Q xy

−Qx−Q y

2 1.1

in order to equip IRⁿwith a structure of anonassociative in generalcommutative algebra A,∗. In the corresponding algebraA,∗the square x∗x x²of each vectorxis equal to

x² Q2x−2Qx

2 2²Qx−2Qx

2 Qx. 1.2

Thus, the system xQxobviously becomes a Riccati equation x x∗x x²and many interesting relations follow.

In the sequel we consider the existence of some special algebraic elements i.e., nilpotents of rank 2 and idempotents, as well as the reflection of algebra isomorphisms in the corresponding homogeneous quadratic systems, which represents the basis for the linear equivalence classification of homogeneous quadratic systems. It was already used by the author in order to analyze the stability of the origin in the continuous case in IR²and in IR³the origin is namely a total degenerated critical point for xQxin any dimensionn 3.

However, in the discrete casex_k1Qx_kthe origin is obviously a super stable critical point, since the Jacobian evaluated at the origin is the zero matrix and consequently it is eigenvalues are all zero. On the other hand the dynamics in discrete systems can readily become chaotic in some special regions of the space even in 1Dcf.4, Section 8and it is well-known5that the dynamics on the unit circlewhich contains the fixed point1,0is chaotic for

x_k1x²_k−y_k²,

yk12xkyk. 1.3

Note that system1.3is a homogeneous quadratici.e., of the formxk1Qxkfor Qx Q

x, y

x²−y²,2xy

. 1.4

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Table 1

∗ e1 e2

e1 e1 e2

e2 e2 −e1

Table 2

∗ 1 i

1 1 i

i i −1

Table 3

∗ e1 e2 e3

e1 e1 e2 e3

e2 e2 −e1 0

e3 e3 0 −e1

The interested reader is invited to consult, for example,6–10to obtain some further informations.

Let us conclude the introduction with two examples in order to explain the one to one connection defined by1.1. Let us consider system1.3and it is continuous analogue:

xx²−y²,y2xy. Their corresponding quadratic form isQx, y x²−y²,2xy. Using 1.1one obtains the following multiplication rule:

x, y

∗u, v

xu−yv, xvyu

. 1.5

Thus, in the standard basise1 1,0ande2 0,1the multiplication table for the corresponding algebra is as illustrated inTable 1.

Applying the substitutioni.e., the algebra isomorphisme1 → 1,e2 → ione obtains as illustrated inTable 2which is readily recognized as the algebra of complex numbers.

On the other hand, beginning, for example, with the algebra A IR³,∗given with the multiplication table as illustrated inTable 3the corresponding quadratic form is obtained again by applyingx∗x x².

By denotingx x, y, z xe1ye2ze3, we get xe1ye2ze3

₂

x²e₁²y²e²₂z²e²₃

2xye1∗e22xze1∗e32yze2∗e3

x²e1−y²e1−z²e12xye22xze3

x²−y²−z²,2xy,2xz Q

x, y, z .

1.6

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Thus, we obtain the following quadratic systems x_k1x_k²−y²_k−z²_k,

yk12xkyk, z_k12x_kz_k, xx²−y²−z²,

y2xy, z2xz.

1.7

2. Some Connections between Systems and Their Corresponding Algebras

First note that the algebra which corresponds to a systemxQxorxk1Qxkis always commutative, since from1.1, it follows

x∗yy∗x; ∀x, y∈A. 2.1

However, the corresponding algebra is generally not associative. For instance for algebra A IR³,∗in the above example fromTable 3one can readily observe

0 e3∗e2∗e2/e3∗e2∗e2 −e3. 2.2 Obviously, the correspondence1.1between system and algebra is unique. Note also that there is a one-to-one correspondence between homogeneous systems of degreemand the corresponding m-ary algebras. In this paper we stay within the domain m 2, but the interested reader is referred to10,11for further informationsin casem >2.

In order to achieve better understanding let us recall some definitions from the dynamical systems and algebra theory. A subset W ⊆ A which is closed for algebraic multiplicationi.e., for every pairw1, w2 ∈W we havew1∗w2 ∈Wis called a subalgebra.

For example, if the corresponding vector space is a direct sum of twovectorsubspacesi.e., V V1⊕V2and ifA1 V1,∗andA2 V2,∗, thenA A1⊕A2 V1⊕V2,∗contains two nontrivial subalgebrasA1 andA2. To everyx ∈A∗ one can associate a subalgebraWx, defined by productsx,x² x∗x,x²∗x,x∗x²,x²∗x∗ x,x∗x²∗x,x²∗x², and so on and their linear combinations, which is called the subalgebra generated by the elementx. A subalgebraI ⊆Ais calledleft and rightideal of algebraA, ifAI ⊆I andIA⊆ Ii.e., for everyi∈I and everyx∈Awe havex∗i∈Iandi∗x∈I. Every algebraA V,∗has at lest two ideals, the trivial idealsV and{0}. Furthermore, the setA² A∗Adefined as the subspace of all linear combinations of products inAis obviously an ideal ofA.

The map φ : A → B is homomorphism from algebra A,∗into algebra B,◦, if and only if, for every pair of vectors x, y from algebraAwe have:φx∗y φx◦φy.

If there is a homomorphism from algebraAto algebra Bthey are called homomorphic. A bijective homomorphism is called an isomorphism and the corresponding algebras are called isomorphicin this case m n. ByS_∗ and S_◦ let us denote the corresponding quadratic

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continuous or discretesystems. The maph : IRⁿ → IRⁿ preserves solutions from system x x∗xinto systemy y◦yif and only if it takes parametrized solutions of the first system into parametrized solutions of the second onei.e.,yt hxtis a solution of systemS◦, wheneverxtis a solution ofS∗. In discrete systems the solutions,xk;k0,1,2, . . .are called orbits. By preserving of orbits we mean thathx_k;k 0,1,2, . . . is an orbit of system S_◦, wheneverx_k;k0,1,2, . . .is an orbit of systemS_∗.

Elementaof algebra A,∗is said to be a nilpotent of rank 2, ifa∗a 0 and it is said to be an idempotent, ifa∗a a. If for some point x0 the algebraic equationQx0 0 or Qx0 x0 is fulfilled, it is called critical point of system x Qx or x_k1 Qx_k, respectively. The solution xtis a ray solution of x Qxif for every timet vector xt remains on the line IRxt.

2.1. Algebraic Isomorphism and Linear Equivalence

The basic correspondence1.1between quadratic systems and algebras is the same forx Qxas well forxk1 Qxk. The basic property concerning the linear equivalence between quadratic systems is also very similar as shown in the following two Propositions.

Proposition 2.1. Letφ : IRⁿ → IR^mbe linear. Thenφ preserves solutions from systemS_∗ : x x∗x;x∈IRⁿinto systemS◦ :y y∗y;y∈IR^mif and only ifφis a homomorphism from algebra A_∗ IRⁿ,∗into algebraA_◦ IR^m,◦.

Proof. Letφ be some linear map which preserves solutions fromS_∗ into S_◦. And letA_∗ IRⁿ,∗andA◦ IR^m,◦be the corresponding algebras. Letxtbe the solution ofS∗and let ytbe the solution ofS_◦. Thusyφxandx x∗xand fromy y◦yone obtainsφx φx ◦ φx. Sinceφ is linear it is Jacobian is equal to φ in every point of the spacei.e., φφ. Thereforeφx φx◦ φxfor everyx∈IRⁿ. Substitutingx XYand applying commutativity and bilinearity of multiplications◦and∗, we obtainφX∗Y φX◦φY, for all X, Y ∈IRⁿ. Sinceφis linear by assumption, this yields thatφis a homomorphism from A_∗ IRⁿ,∗intoA_◦ IR^m,◦.

Conversely, letφbe homomorphism fromA_∗ IRⁿ,∗intoA_◦ IR^m,◦. Thus, for all X, Y ∈ IRⁿ we haveφX∗ Y φX◦φY. For X Y we readily obtain:φX∗ X φX◦φX. Using againφφ, we obtain

φ X∗ X

φ X

◦φ X

. 2.3

Let Xtbe a solution ofS_∗. We want to prove thatφXis a solution ofS_◦. Using X X∗ X and2.3and the chain rule for the derivative one obtains φX φX· X φX φX◦φX, which means thatφXis a solution ofS_◦. This completes the proof.

Proposition 2.2. Letφ : IRⁿ → IR^m be linear. Thenφpreserves orbits from systemS∗ : xk1 xk∗xk;x∈IRⁿinto systemS◦ : yk1 yk∗yk;y∈IR^mif and only ifφis a homomorphism from algebraA_∗ IRⁿ,∗into algebraA_◦ IR^m,◦.

The proof is very similar to the proof ofProposition 2.1and will be omitted here.

The use of Propositions2.1and2.2is quite similar. In the following Example the use ofProposition 2.1is considered.

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−6 −4 −2 0 2

−2 0 2 4 6 8

Figure 1: The particular solution toSX,Y.

Example 2.3. Systems

S_x,y: xx²−y² y2xy,

S_X,Y: X−62X²−100XY −40Y² Y85X²136XY 54Y²

2.4

are isomorphic. The corresponding isomorphism fromx, yintoX, Yis

Φ

⎡

⎣ 2 −1

−5 2

3 2

⎤

⎦

−1

. 2.5

Note that systemS_x,y is much easier to treat than S_X,Y. The only idempotent ofS_x,y is x1,y0, while the only idempotent ofS_X,YisX 2,Y −5/2. It is obtained as the solution to algebraic system of equations

X−62X²−100XY−40Y²

Y 85X²136XY54Y². 2.6

The particular solutions with the initial conditions near idempotentthe black linein both cases yield the solution curvesthe red lineshown in Figures1and2. Figures1,2,3, and4 are clearly indicating that the dynamics of systemSx,yis much easier to understand. Note that in the Markus theory systemSx,yis a kind of normal formi.e., the class representative of it is classi.e., of all isomorphic systems. For the entire list of “normal forms” in 2D please refer to2, Theorems 6, 7, and 8.

The immediate corollary is that systemsS∗andS◦are linearly equivalent if and only if their corresponding algebrasA_∗andA_◦are isomorphic.

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Figure 2: The particular solution toSx,y.

−5 −4 −3 −2 −1 0 1 2 3 4 X5

−5

−4

−3

−2

−1 0 1 2 3 4 Y5

Figure 3: The phase diagram basis forSx,y.

2.2. Algebraic Structure and Reductions of the System

The above-mentioned corollary and the so-called Kaplan-Yorke theorem is a basement of algebraic treatment of homogeneous quadratic systems using algebraic classification of the commutative algebras. The following algebraic result due by Kaplan and Yorke12aﬀects strongly on the dynamics of homogeneous quadratic systems.

Theorem 2.4Kaplan-Yorke. Every real finite dimensional algebraA_∗ IR^m,∗contains at least one nonzero idempotent or a nonzero nilpotent of rank two.

For proof please refer to the original paper12.

Concerning the existence of a subalgebra, we have the following result.

Proposition 2.5. A homogeneous quadratic systemS∗has an invariantr-dimensional linear subspace E_rif and only if the corresponding algebra has anr-dimensional subalgebra.

Remark 2.6. We present just the proof for discrete casei.e., whenS∗: xk1 xk∗xk;x∈IRⁿ. The proof for continuous systemS∗:x x∗x;x∈IRⁿcan be found, for example, in Markus 2.

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−5 −4 −3 −2 −1 0 1 2 3 4 X5

−5

−4

−3

−2

−1 0 1 2 3 4 Y 5

Figure 4: The phase diagram basis forSX,Y.

Proof. Let Er spane1, e2, . . . , er be an invariant r-dimensional linear subspace of a n- dimensional vector spaceV,n > r. Then for everyx∈E_r the orbit{x, x² x∗x, x²∗x, x∗ x², x²∗ x²,x²∗ x∗ x,x∗ x²∗ x, . . .}is contained inE_r. We will prove thatE_r,∗is the r-dimensional subalgebrai.e., the subspaceEr is closed for multiplication∗. Settingxei

we haveei∗ei ∈Er for every 1≤i≤r. Now we want to prove thatei∗ej ∈Er for all 1≤i, j≤r. In order to prove this, let us setxe_ie_j ∈E_rand computex²using the commutativity rule in algebra

x² eiej

∗ eiej

ei∗ei2ei∗ejej∗ej. 2.7

Sincex²∈Er,ei∗ei∈Er, andej∗ej∈Erit follows alsoei∗ej∈Erwhich means that for every pairx, y∈E_rthe productx∗yis contained inE_r, as stated. The converse follows directly from the fact that for everyx, y∈E_r, sinceE_ris a subalgebra, we havex∗y∈E_r. Settingyxx0

we immediately obtain thatx0∗x0∈Er. Settingyxx0∗x0one obtainsx0∗x0∗x0∗x0∈ E_r, and so on. Thus the orbit{x, x²x∗x, x²∗x, x∗x², x²∗x²,x²∗x∗x,x∗x²∗x, . . .}is contained inE_r, which means thatE_r is invariant, as stated.

Concerning the existence of a subalgebra and an ideal in the corresponding algebra let us mention the following result,for proof please refer to10.

Proposition 2.7. LetIbe an ideal of algebraA_∗andWa subagebra such thatV I⊕W. Then the solution of the initial value problem of the corresponding quadratic system x x∗xwith the initial value problemx0 x0 w0i0can be solved by successive solution of

ww∗w; w0 w0 inW,

ii∗i2wt∗i; i0 i0 inI, 2.8

wherewtis a solution of the first subsystem inW.

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Corollary 2.8. A system x x∗ xwith the initial condition x0 x0 splits into two separated subsystems

ww∗w; w0 w0 inW,

ii∗i; i0 i0 in I, 2.9

if and only if the corresponding algebra can be written as a direct sum of two nontrivial ideals

AI1⊕I2. 2.10

Proof. ApplyI I1andW I2 in the previous result and take into consideration thatI1, I2

are both ideals which means that 2wt∗i0 in the second equation ofProposition 2.7. This finishes the proof.

The last two results are further examples where exactly analogous results can be formulated for the discrete case. Note that the reduction and/or splitting of the system is of great importance when exact solutions are needed.

2.3. Special Algebraic Elements and (In)Stability

However, some connections between the system and corresponding algebra A,∗ diﬀers in the continuous and discrete case. For example the correspondence between ray solutions/fixed points and idempotents/nilpotents. Let us first recall the Lyapunov definition of stability.

Definition 2.9. Critical pointx0of systemx x∗xis said to be stable if and only if for every ε >0 there is aδ >0 such that for every initial conditionx0for whichx0< δand for every timet >0 for which the solutionxx0, twith the initial conditionx0is defined, we have

xx0, t< ε. 2.11

In the next theorem the well-known necessary conditions for the stability of the origin in x x∗xare given.

Theorem 2.10. If an algebra A_∗ contains an idempotent p ∗ p p, then the origin in the corresponding systemxx∗xis unstable critical point.

Proof. First note that IR·pis always a subalgebra ofA∗. ThusbyProposition 2.5, the flow ftp is invariant. Sincep² p from x x∗ xwhen inserting xt ftpone obtains 1-dimensional ODEft f²t. Next observe thatεpis in every neighborhood of the origin.

Therefore the solution with the initial conditionx0εpi.e.,f0 εis

xt ε

1−εt ·p; fort∈

0,1 ε

. 2.12

Finally observe that lim_t_→1/εxt∞which completes the proof.

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Note that the immediate corollary ofTheorem 2.10and the Kaplan-Yorke theorem is that systems x x∗xwith the stable origin always contain some nilpotents of rank two. In the continuous case the ray-solutions are as proven inTheorem 2.10related with the existence of the idempotent. However, in the discrete case the existence of idempotent simply means the existence of the fixed point.

On the other hand, the existence of a nilpotentnof rank two implies the existence of line of critical points IR·nin the continuous case, since fromn∗n0 one obtainsαn∗αn α²00 for every realα. However, in the discrete case the above property yields the existence of the ray-solution, since fromx_k αnone readily obtains thatx_k1 αn∗αn α²00 for every realα.

3. Conclusions

For the stability analysis of the origin in systemsxx∗xsome new results are needed, for example, results obtained Markus approach in13,14. Using Markus original classification one can obtain that onlyup to linear equivalencethreefamilies ofsystems admit stable origin in 2D. These systems arecf.13:

x0, x−y², xky², k <−1/8,

y0, y2xy, y2xyy². 3.1

In order to obtain similar resultsin IR³ and/or in IRⁿforn > 3 a partial algebraic classificationof systems/algebras with a plane of critical pointssimilar to Markus was done incf.14. Roughly speakingcf.13, the existence of complex idempotents overlapping with the existence of the so-called essential nilpotentsi.e., nilpotents which are not contained in the linear span of all complex idempotentsseem to definealgebraicallythe stability of the origin. The conjecture was confirmed by examining the complexification

C∗:A∗iA∗ 3.2

of real algebrasA_∗corresponding to the systems with a plane of critical points as well as on the so-called homogenized systems in IR³cf.15. It seems that13the spectral analysis of linear operatorLndefined byLna : a∗ni.e., multiplication by essential nilpotentnis playing an important role in stability of the origin in systemsx x∗x.

However, algebraic approach is recently used cf. 8,9 also in order to consider planar homogeneous discrete systems in the sense ofnonchaotic dynamics. The results are showing that the dynamics of systems whose corresponding algebras are containing some nilpotents of rank 2 cannot be chaotic9. Furthermore, system1.3is one of the simplest systems with chaotic dynamics and the corresponding algebra A2 is power-associative.

Note that every orbit of system xk1 xk∗ xk which corresponds to a power-associative algebra can be obtained in terms of an orbit of a corresponding linear system. Namely, given an initial point x0 the orbit of xk1 xk∗ xk can be obtained in terms of xk1 Lx0xk, since in the power-associative algebras the powers of everyx0 are well definedi.e., xⁿ¹₀ x0∗xⁿ₀ x₀²∗xⁿ⁻¹₀ · · ·. In case of system1.3the left multiplication matrix ofx x, y by

X X, Yis obtained fromx∗ X xX−yY, xY yX _x_−y

y x _X

Y

.

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In the chaotic region wherex 1, the corresponding multiplication matrix has the form

Lcosφ,sinφ

cosφ −sinφ sinφ cosφ

. 3.3

Readily, ifφ kπwherek is a rational number, then the pointcosφ,sinφis periodic. On the other hand, ifφKπwhereKis irrational, the orbit ofcosφ,sinφis dense on the unit circlex1 but not periodic. Furthermore, the pointscosφ,sinφ whereφ kπandkis a rational numberare dense inx1, as well. Thus there is chaos onx1. The question is whether the other power-associative algebras also correspond to the systems with chaotic dynamics.

Finally, note that in the continuous case one can observe the following: the solution to x x∗ xwith the initial condition x0 x0 can be expressed explicitly by the following formula:

xt I−tL_x₀x0⁻¹x0, 3.4

whereIis the identity matrix andLx0is the linear operator defined by theleftmultiplication byx0. The proof of the above explicit formula is a direct computation and can be found in 7, where3.4is used to prove that in power-associative algebras the corresponding system x x∗ xcannot have periodic solutions. Another interesting questionwhen considering power-associativity together with continuous quadratic systemsis whether one can use3.4 in order to obtain some results on stability of the origin in IRⁿn≥2.

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