# On Algebraic Approach in Quadratic Systems

Authors: Matej Mencinger
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

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 (non)chaotic 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 in (1960). We resume some connections between the dynamics of the quadratic systems and (algebraic) properties of the corresponding algebras. We consider some general connections and the influence of power-associativity in the corresponding quadratic system.


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## Body

## 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 ofx⃗′=f(x⃗) or x⃗k+1=f(x⃗k) is defined to be the solution of the following algebraic (system of) equation(s), f(x⃗0)=0⃗ or f(x⃗0)=x⃗0, respectively. For the systems of ODEs, x⃗′=f(x⃗), the critical point x⃗0 is said to be hyperbolic if no eigenvalue of the corresponding Jacobian matrix, Jf(x⃗0), of the (nonlinear vector) function f has it is eigenvalue equal to zero (i.e., Re(λi)≠0). In case of discrete system, x⃗k+1=f(x⃗k), the critical point x⃗0 is said to be hyperbolic if no eigenvalue of the Jacobian matrix has it is eigenvalue equal to 1 (i.e., |λi|≠1). Roughly speaking, if for a continuous system Re(λi)<0 for every λi, the corresponding critical point is stable (it is unstable, if Re(λi)>0 for some λi). Similar, if for discrete systems |λi|<1 for every λi, the corresponding critical point is stable (it is unstable, if |λi|>1 for some λi). Note that just one eigenvalue of the corresponding linear approximation of x⃗′=f(x⃗) or x⃗k+1=f(x⃗k) for which Re(λi)=0 or |λi|=1, respectively, implies that the stability must be investigated separately in each particular case (because 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 be [1]. 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 formx⃗′=Q(x⃗) and autonomous homogeneous quadratic discrete dynamical systems of the form x⃗k+1=Q(x⃗k) (where Q:IRn→IRn is homogeneous of degree two in each component: Q(ax⃗)=a2Q(x⃗) for each (real) a) is considered, as suggested by Markus in [2]. Markus idea was to define a unique algebra multiplication via the following bilinear form B(x⃗,y⃗)=x⃗*y⃗:
(1.1)x⃗*y⃗:=Q(x⃗+y⃗)-Q(x⃗)-Q(y⃗)2
in order to equip IRn with a structure of a (nonassociative in general) commutative algebra (A,*). In the corresponding algebra (A,*) the square x⃗*x⃗=x⃗2 of each vector x⃗ is equal to
(1.2)x⃗2=Q(2x⃗)-2Q(x⃗)2=22Q(x⃗)-2Q(x⃗)2=Q(x⃗).
Thus, the system x⃗′=Q(x⃗) obviously becomes a Riccati equation x⃗′=x⃗*x⃗=x⃗2 and 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 inIR2 and in IR3 (the origin is namely a total degenerated critical point for x⃗′=Q(x⃗) in any dimension n [3]).However, in the discrete casex⃗k+1=Q(x⃗k) the 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 1D (cf. [4, Section 8]) and it is well-known [5] that the dynamics on the unit circle (which contains the fixed point (1,0) is chaotic for(1.3)xk+1=xk2-yk2,yk+1=2xkyk.Note that system (1.3) is a homogeneous quadratic (i.e., of the form x⃗k+1=Q(x⃗k)) for(1.4)Q(x⃗)=Q(x,y)=(x2-y2,2xy).The interested reader is invited to consult, for example, [6–10] to obtain some further informations.Let us conclude the introduction with two examples in order to explain the one to one connection defined by (1.1). Let us consider system (1.3) and it is continuous analogue: x′=x2-y2, y′=2xy. Their corresponding quadratic form is Q(x,y)=(x2-y2,2xy). Using (1.1) one obtains the following multiplication rule:(1.5)(x,y)*(u,v)=(xu-yv,xv+yu).Thus, in the standard basise1=(1,0) and e2=(0,1) the multiplication table for the corresponding algebra is as illustrated in Table 1.Table 1*e1e2e1e1e2e2e2-e1Applying the substitution (i.e., the algebra isomorphism)e1→1, e2→i one obtains as illustrated in Table 2 which is readily recognized as the algebra of complex numbers.Table 2*1i11iii-1On the other hand, beginning, for example, with the algebraA=(IR3,*) given with the multiplication table as illustrated in Table 3 the corresponding quadratic form is obtained again by applying x⃗*x⃗=x⃗2.Table 3*e1e2e3e1e1e2e3e2e2-e10e3e30-e1By denotingx⃗=(x,y,z)=xe1+ye2+ze3, we get(1.6)(xe1+ye2+ze3)2=x2e12+y2e22+z2e32+2xye1*e2+2xze1*e3+2yze2*e3=x2e1-y2e1-z2e1+2xye2+2xze3=(x2-y2-z2,2xy,2xz)=Q(x,y,z).
Thus, we obtain the following quadratic systems
(1.7)xk+1=xk2-yk2-zk2,yk+1=2xkyk,zk+1=2xkzk,x′=x2-y2-z2,y′=2xy,z′=2xz.


## 2. Some Connections between Systems and Their Corresponding Algebras

First note that the algebra which corresponds to a systemx⃗′=Q(x⃗) or x⃗k+1=Q(x⃗k) is always commutative, since from (1.1), it follows
(2.1)x⃗*y⃗=y⃗*x⃗;∀x⃗,y⃗∈A.
However, the corresponding algebra is generally not associative. For instance for algebra A=(IR3,*) in the above example from Table 3 one can readily observe
(2.2)0=(e3*e2)*e2≠e3*(e2*e2)=-e3.
Obviously, the correspondence (1.1) between system and algebra is unique. Note also that there is a one-to-one correspondence between homogeneous systems of degree m and the corresponding m-ary algebras. In this paper we stay within the domain m=2, but the interested reader is referred to [10, 11] for further informations (in case m>2).In order to achieve better understanding let us recall some definitions from the dynamical systems and algebra theory. A subsetW⊆A which is closed for algebraic multiplication (i.e., for every pair w1,w2∈W we have w1*w2∈W) is called a subalgebra. For example, if the corresponding vector space is a direct sum of two (vector) subspaces (i.e., V=V1⊕V2) and if A1=(V1,*) and A2=(V2,*), then A=A1⊕A2=(V1⊕V2,*) contains two nontrivial subalgebras A1 and A2. To every x⃗∈A* one can associate a subalgebra Wx⃗, defined by products x⃗, x⃗2=x⃗*x⃗, x⃗2*x⃗, x⃗*x⃗2, (x⃗2*x⃗)*x⃗, (x⃗*x⃗2)*x⃗, x⃗2*x⃗2, and so on and their linear combinations, which is called the subalgebra generated by the element x⃗. A subalgebra I⊆A is called (left and right) ideal of algebra A, if AI⊆I and IA⊆I (i.e., for every i∈I and every x⃗∈A we have x⃗*i∈I and i*x⃗∈I). Every algebra A=(V,*) has at lest two ideals, the trivial ideals V and {0}. Furthermore, the set A2=A*A defined as the subspace of all linear combinations of products in A is obviously an ideal of A.The mapϕ:A→B is homomorphism from algebra (A,*) into algebra (B,∘), if and only if, for every pair of vectors x⃗,y⃗ from algebra A we have: ϕ(x⃗*y⃗)=ϕ(x⃗)∘ϕ(y⃗). If there is a homomorphism from algebra A to algebra B they are called homomorphic. A bijective homomorphism is called an isomorphism and the corresponding algebras are called isomorphic (in this case m=n). By S* and S∘ let us denote the corresponding quadratic (continuous or discrete) systems. The map h:IRn→IRn preserves solutions from system x⃗′=x⃗*x⃗ into system y⃗′=y⃗∘y⃗ if and only if it takes parametrized solutions of the first system into parametrized solutions of the second one (i.e., y⃗(t)=h(x⃗(t)) is a solution of system S∘, whenever x⃗(t) is a solution of S*. In discrete systems the solutions, x⃗k; k=0,1,2,… are called orbits. By preserving of orbits we mean that h(x⃗k); k=0,1,2,… is an orbit of system S∘, whenever x⃗k; k=0,1,2,… is an orbit of system S*.Elementa⃗ of algebra (A,*) is said to be a nilpotent of rank 2, if a⃗*a⃗=0⃗ and it is said to be an idempotent, if a⃗*a⃗=a⃗. If for some point x⃗0 the algebraic equation Q(x⃗0)=0⃗ or Q(x⃗0)=x⃗0 is fulfilled, it is called critical point of system x⃗′=Q(x⃗) or x⃗k+1=Q(x⃗k), respectively. The solution x⃗(t) is a ray solution of x⃗′=Q(x⃗) if for every time t vector x⃗(t) remains on the line IRx⃗(t).


### 2.1. Algebraic Isomorphism and Linear Equivalence

The basic correspondence (1.1) between quadratic systems and algebras is the same for x⃗′=Q(x⃗) as well for x⃗k+1=Q(x⃗k). 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→IRm be linear. Then ϕ preserves solutions from system S*:x⃗′=x⃗*x⃗; x⃗∈IRn into system S∘:y⃗′=y⃗*y⃗; y⃗∈IRm if and only if ϕ is a homomorphism from algebra A*=(IRn,*) into algebra A∘=(IRm,∘).Proof.
Letϕ be some linear map which preserves solutions from S* into S∘. And let A*=(IRn,*) and A∘=(IRm,∘) be the corresponding algebras. Let x⃗(t) be the solution of S* and let y⃗(t) be the solution of S∘. Thus y⃗=ϕ(x⃗) and x⃗′=x⃗*x⃗ and from y⃗′=y⃗∘y⃗ one obtains (ϕ(x⃗))′=ϕ(x⃗)∘ϕ(x⃗). Since ϕ is linear it is Jacobian is equal to ϕ in every point of the space (i.e., ϕ′=ϕ). Therefore ϕ(x⃗)=ϕ(x⃗)∘ϕ(x⃗) for every x⃗∈IRn. Substituting x⃗=X⃗+Y⃗ and applying commutativity and bilinearity of multiplications ∘ and *, we obtain ϕ(X⃗*Y⃗)=ϕ(X⃗)∘ϕ(Y⃗), for all X⃗,Y⃗∈IRn. Since ϕ is linear by assumption, this yields that ϕ is a homomorphism from A*=(IRn,*) into A∘=(IRm,∘).
Conversely, letϕ be homomorphism from A*=(IRn,*) into A∘=(IRm,∘). Thus, for all X⃗,Y⃗∈IRn we have ϕ(X⃗*Y⃗)=ϕ(X⃗)∘ϕ(Y⃗). For X⃗=Y⃗ we readily obtain: ϕ(X⃗*X⃗)=ϕ(X⃗)∘ϕ(X⃗). Using again ϕ′=ϕ, we obtain(2.3)ϕ′(X⃗*X⃗)=ϕ(X⃗)∘ϕ(X⃗).
Let X⃗(t) be a solution of S*. We want to prove that ϕ(X⃗) is a solution of S∘. Using X⃗′=X⃗*X⃗ and (2.3) and the chain rule for the derivative one obtains ϕ′(X⃗′)=ϕ′(X⃗)·X⃗′=(ϕ(X⃗))′=ϕ(X⃗)∘ϕ(X⃗), which means that ϕ(X⃗) is a solution of S∘. This completes the proof.Proposition 2.2.
Letϕ:IRn→IRm be linear. Then ϕ preserves orbits from system S*:x⃗k+1=x⃗k*x⃗k; x⃗∈IRn into system S∘:y⃗k+1=y⃗k*y⃗k; y⃗∈IRm if and only if ϕ is a homomorphism from algebra A*=(IRn,*) into algebra A∘=(IRm,∘).The proof is very similar to the proof of Proposition2.1 and will be omitted here.The use of Propositions2.1 and 2.2 is quite similar. In the following Example the use of Proposition 2.1 is considered.Example 2.3.
Systems(2.4)S(x,y):x′=x2-y2y′=2xy,S(X,Y):X′=-62X2-100XY-40Y2Y′=85X2+136XY+54Y2
are  isomorphic. The corresponding isomorphism from (x,y) into (X,Y) is
(2.5)Φ=[2-1-5232]-1.
Note that system S(x,y) is much easier to treat than S(X,Y). The only idempotent of S(x,y) is (x=1, y=0),while the only idempotent of S(X,Y) is (X=2, Y=-5/2). It is obtained as the solution to algebraic system of equations
(2.6)X=-62X2-100XY-40Y2Y=85X2+136XY+54Y2.
The particular solutions with the initial conditions near idempotent (the black line) in both cases yield the solution curves (the red line) shown in Figures 1 and 2. Figures 1, 2, 3, and 4 are clearly indicating that the dynamics of system S(x,y) is much easier to understand. Note that in the Markus theory system S(x,y) is a kind of normal form (i.e., the class representative) of it is class (i.e., of all isomorphic systems). For the entire list of “normal forms” in 2D please refer to [2, Theorems 6, 7, and 8].Figure 1
The particular solution toS(X,Y).Figure 2
The particular solution toS(x,y).Figure 3
The phase diagram basis forS(x,y).Figure 4
The phase diagram basis forS(X,Y).The immediate corollary is that systemsS* and S∘ are linearly equivalent if and only if their corresponding algebras A* and A∘ are isomorphic.


### 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 Yorke [12] affects strongly on the dynamics of homogeneous quadratic systems.Theorem 2.4 (Kaplan-Yorke).
Every real finite dimensional algebraA*=(IRm,*) contains at least one nonzero idempotent or a nonzero nilpotent of rank two.For proof please refer to the original paper [12].Concerning the existence of a subalgebra, we have the following result.Proposition 2.5.
A homogeneous quadratic systemS* has an invariant r-dimensional linear subspace Er if and only if the corresponding algebra has an r-dimensional subalgebra.Remark 2.6.
We present just the proof for discrete case (i.e., whenS*:x⃗k+1=x⃗k*x⃗k; x⃗∈IRn. The proof for continuous system S*:x⃗′=x⃗*x⃗; x⃗∈IRn can be found, for example, in Markus [2].Proof.
LetEr=span(e1,e2,…,er) be an invariant r-dimensional linear subspace of a n-dimensional vector space V, n>r. Then for every x⃗∈Er the orbit {x⃗,x⃗2=x⃗*x⃗,x⃗2*x⃗,x⃗*x⃗2,x⃗2*x⃗2,(x⃗2*x⃗)*x⃗,(x⃗*x⃗2)*x⃗,…} is contained in Er. We will prove that (Er,*) is the r-dimensional subalgebra (i.e., the subspace Er is closed for multiplication *). Setting x⃗=ei we have ei*ei∈Er for every 1≤i≤r. Now we want to prove that ei*ej∈Er for all 1≤i, j≤r. In order to prove this, let us set x⃗=ei+ej∈Er and compute x⃗2 using the commutativity rule in algebra
(2.7)x⃗2=(ei+ej)*(ei+ej)=ei*ei+2ei*ej+ej*ej.
Since x⃗2∈Er, ei*ei∈Er, and ej*ej∈Er it follows also ei*ej∈Er which means that for every pair x⃗,y⃗∈Er the product x⃗*y⃗ is contained in Er, as stated. The converse follows directly from the fact that for every x⃗,y⃗∈Er, since Er is a subalgebra, we have x⃗*y⃗∈Er. Setting y⃗=x⃗=x⃗0 we immediately obtain that x⃗0*x⃗0∈Er. Setting y⃗=x⃗=x⃗0*x⃗0 one obtains (x⃗0*x⃗0)*(x⃗0*x⃗0)∈Er, and so on. Thus the orbit {x⃗,x⃗2=x⃗*x⃗,x⃗2*x⃗,x⃗*x⃗2,x⃗2*x⃗2,(x⃗2*x⃗)*x⃗,(x⃗*x⃗2)*x⃗,…} is contained in Er, which means that Er 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 to [10]).Proposition 2.7.
LetI be an ideal of algebra A* and W a subagebra such that V=I⊕W. Then the solution of the initial value problem of the corresponding quadratic system x⃗′=x⃗*x⃗ with the initial value problem x⃗(0)=x⃗0=w0+i0 can be solved by successive solution of
(2.8)w′=w*w;w(0)=w0inW,i′=i*i+2w(t)*i;i(0)=i0inI,
where w(t) is a solution of the first subsystem in W.Corollary 2.8.
A systemx⃗′=x⃗*x⃗ with the initial condition x⃗(0)=x⃗0 splits into two separated subsystems
(2.9)w′=w*w;w(0)=w0inW,i′=i*i;i(0)=i0inI,
if and only if the corresponding algebra can be written as a direct sum of two nontrivial ideals
(2.10)A=I1⊕I2.Proof.
ApplyI=I1 and W=I2 in the previous result and take into consideration that I1,I2 are both ideals which means that 2w(t)*i=0 in the second equation of Proposition 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 pointx⃗0 of system x⃗′=x⃗*x⃗ is said to be stable if and only if for every ε>0 there is a δ>0 such that for every initial condition x⃗0 for which ∥x⃗0∥<δ and for every time t>0 for which the solution x⃗(x⃗0,t) with the initial condition x⃗0 is defined, we have
(2.11)‖x⃗(x⃗0,t)‖<ε.
In the next theorem the well-known necessary conditions for the stability of the origin in x⃗′=x⃗*x⃗ are given.Theorem 2.10.
If an algebraA* contains an idempotent p⃗*p⃗=p⃗, then the origin in the corresponding system x⃗′=x⃗*x⃗ is unstable critical point.Proof.
First note thatIR·p⃗ is always a subalgebra of A*. Thus (by Proposition 2.5), the flow f(t)p⃗ is invariant. Since p⃗2=p⃗ from x⃗′=x⃗*x⃗ (when inserting x⃗(t)=f(t)p⃗) one obtains 1-dimensional ODE f′(t)=f2(t). Next observe that εp⃗ is in every neighborhood of the origin. Therefore the solution with the initial condition x⃗0=εp⃗ (i.e., f(0)=ε) is
(2.12)x⃗(t)=ε1-εt⋅p⃗;fort∈[0,1ε).
Finally observe that limt→1/ε∥x⃗(t)∥=∞ which completes the proof.Note that the immediate corollary of Theorem2.10 and the Kaplan-Yorke theorem is that systems x⃗′=x⃗*x⃗ with the stable origin always contain some nilpotents of rank two. In the continuous case the ray-solutions are as proven in Theorem 2.10 related 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 nilpotentn⃗ of rank two implies the existence of line of critical points IR·n⃗ in the continuous case, since from n⃗*n⃗=0⃗ one obtains (αn⃗)*(αn⃗)=α20⃗=0⃗ for every realα. However, in the discrete case the above property yields the existence of the ray-solution, since from x⃗k=αn⃗ one readily obtains that x⃗k+1=(αn⃗)*(αn⃗)=α20⃗=0⃗ for every real α.


## 2.1. Algebraic Isomorphism and Linear Equivalence

The basic correspondence (1.1) between quadratic systems and algebras is the same for x⃗′=Q(x⃗) as well for x⃗k+1=Q(x⃗k). 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→IRm be linear. Then ϕ preserves solutions from system S*:x⃗′=x⃗*x⃗; x⃗∈IRn into system S∘:y⃗′=y⃗*y⃗; y⃗∈IRm if and only if ϕ is a homomorphism from algebra A*=(IRn,*) into algebra A∘=(IRm,∘).Proof.
Letϕ be some linear map which preserves solutions from S* into S∘. And let A*=(IRn,*) and A∘=(IRm,∘) be the corresponding algebras. Let x⃗(t) be the solution of S* and let y⃗(t) be the solution of S∘. Thus y⃗=ϕ(x⃗) and x⃗′=x⃗*x⃗ and from y⃗′=y⃗∘y⃗ one obtains (ϕ(x⃗))′=ϕ(x⃗)∘ϕ(x⃗). Since ϕ is linear it is Jacobian is equal to ϕ in every point of the space (i.e., ϕ′=ϕ). Therefore ϕ(x⃗)=ϕ(x⃗)∘ϕ(x⃗) for every x⃗∈IRn. Substituting x⃗=X⃗+Y⃗ and applying commutativity and bilinearity of multiplications ∘ and *, we obtain ϕ(X⃗*Y⃗)=ϕ(X⃗)∘ϕ(Y⃗), for all X⃗,Y⃗∈IRn. Since ϕ is linear by assumption, this yields that ϕ is a homomorphism from A*=(IRn,*) into A∘=(IRm,∘).
Conversely, letϕ be homomorphism from A*=(IRn,*) into A∘=(IRm,∘). Thus, for all X⃗,Y⃗∈IRn we have ϕ(X⃗*Y⃗)=ϕ(X⃗)∘ϕ(Y⃗). For X⃗=Y⃗ we readily obtain: ϕ(X⃗*X⃗)=ϕ(X⃗)∘ϕ(X⃗). Using again ϕ′=ϕ, we obtain(2.3)ϕ′(X⃗*X⃗)=ϕ(X⃗)∘ϕ(X⃗).
Let X⃗(t) be a solution of S*. We want to prove that ϕ(X⃗) is a solution of S∘. Using X⃗′=X⃗*X⃗ and (2.3) and the chain rule for the derivative one obtains ϕ′(X⃗′)=ϕ′(X⃗)·X⃗′=(ϕ(X⃗))′=ϕ(X⃗)∘ϕ(X⃗), which means that ϕ(X⃗) is a solution of S∘. This completes the proof.Proposition 2.2.
Letϕ:IRn→IRm be linear. Then ϕ preserves orbits from system S*:x⃗k+1=x⃗k*x⃗k; x⃗∈IRn into system S∘:y⃗k+1=y⃗k*y⃗k; y⃗∈IRm if and only if ϕ is a homomorphism from algebra A*=(IRn,*) into algebra A∘=(IRm,∘).The proof is very similar to the proof of Proposition2.1 and will be omitted here.The use of Propositions2.1 and 2.2 is quite similar. In the following Example the use of Proposition 2.1 is considered.Example 2.3.
Systems(2.4)S(x,y):x′=x2-y2y′=2xy,S(X,Y):X′=-62X2-100XY-40Y2Y′=85X2+136XY+54Y2
are  isomorphic. The corresponding isomorphism from (x,y) into (X,Y) is
(2.5)Φ=[2-1-5232]-1.
Note that system S(x,y) is much easier to treat than S(X,Y). The only idempotent of S(x,y) is (x=1, y=0),while the only idempotent of S(X,Y) is (X=2, Y=-5/2). It is obtained as the solution to algebraic system of equations
(2.6)X=-62X2-100XY-40Y2Y=85X2+136XY+54Y2.
The particular solutions with the initial conditions near idempotent (the black line) in both cases yield the solution curves (the red line) shown in Figures 1 and 2. Figures 1, 2, 3, and 4 are clearly indicating that the dynamics of system S(x,y) is much easier to understand. Note that in the Markus theory system S(x,y) is a kind of normal form (i.e., the class representative) of it is class (i.e., of all isomorphic systems). For the entire list of “normal forms” in 2D please refer to [2, Theorems 6, 7, and 8].Figure 1
The particular solution toS(X,Y).Figure 2
The particular solution toS(x,y).Figure 3
The phase diagram basis forS(x,y).Figure 4
The phase diagram basis forS(X,Y).The immediate corollary is that systemsS* and S∘ are linearly equivalent if and only if their corresponding algebras A* and A∘ are isomorphic.


## 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 Yorke [12] affects strongly on the dynamics of homogeneous quadratic systems.Theorem 2.4 (Kaplan-Yorke).
Every real finite dimensional algebraA*=(IRm,*) contains at least one nonzero idempotent or a nonzero nilpotent of rank two.For proof please refer to the original paper [12].Concerning the existence of a subalgebra, we have the following result.Proposition 2.5.
A homogeneous quadratic systemS* has an invariant r-dimensional linear subspace Er if and only if the corresponding algebra has an r-dimensional subalgebra.Remark 2.6.
We present just the proof for discrete case (i.e., whenS*:x⃗k+1=x⃗k*x⃗k; x⃗∈IRn. The proof for continuous system S*:x⃗′=x⃗*x⃗; x⃗∈IRn can be found, for example, in Markus [2].Proof.
LetEr=span(e1,e2,…,er) be an invariant r-dimensional linear subspace of a n-dimensional vector space V, n>r. Then for every x⃗∈Er the orbit {x⃗,x⃗2=x⃗*x⃗,x⃗2*x⃗,x⃗*x⃗2,x⃗2*x⃗2,(x⃗2*x⃗)*x⃗,(x⃗*x⃗2)*x⃗,…} is contained in Er. We will prove that (Er,*) is the r-dimensional subalgebra (i.e., the subspace Er is closed for multiplication *). Setting x⃗=ei we have ei*ei∈Er for every 1≤i≤r. Now we want to prove that ei*ej∈Er for all 1≤i, j≤r. In order to prove this, let us set x⃗=ei+ej∈Er and compute x⃗2 using the commutativity rule in algebra
(2.7)x⃗2=(ei+ej)*(ei+ej)=ei*ei+2ei*ej+ej*ej.
Since x⃗2∈Er, ei*ei∈Er, and ej*ej∈Er it follows also ei*ej∈Er which means that for every pair x⃗,y⃗∈Er the product x⃗*y⃗ is contained in Er, as stated. The converse follows directly from the fact that for every x⃗,y⃗∈Er, since Er is a subalgebra, we have x⃗*y⃗∈Er. Setting y⃗=x⃗=x⃗0 we immediately obtain that x⃗0*x⃗0∈Er. Setting y⃗=x⃗=x⃗0*x⃗0 one obtains (x⃗0*x⃗0)*(x⃗0*x⃗0)∈Er, and so on. Thus the orbit {x⃗,x⃗2=x⃗*x⃗,x⃗2*x⃗,x⃗*x⃗2,x⃗2*x⃗2,(x⃗2*x⃗)*x⃗,(x⃗*x⃗2)*x⃗,…} is contained in Er, which means that Er 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 to [10]).Proposition 2.7.
LetI be an ideal of algebra A* and W a subagebra such that V=I⊕W. Then the solution of the initial value problem of the corresponding quadratic system x⃗′=x⃗*x⃗ with the initial value problem x⃗(0)=x⃗0=w0+i0 can be solved by successive solution of
(2.8)w′=w*w;w(0)=w0inW,i′=i*i+2w(t)*i;i(0)=i0inI,
where w(t) is a solution of the first subsystem in W.Corollary 2.8.
A systemx⃗′=x⃗*x⃗ with the initial condition x⃗(0)=x⃗0 splits into two separated subsystems
(2.9)w′=w*w;w(0)=w0inW,i′=i*i;i(0)=i0inI,
if and only if the corresponding algebra can be written as a direct sum of two nontrivial ideals
(2.10)A=I1⊕I2.Proof.
ApplyI=I1 and W=I2 in the previous result and take into consideration that I1,I2 are both ideals which means that 2w(t)*i=0 in the second equation of Proposition 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 pointx⃗0 of system x⃗′=x⃗*x⃗ is said to be stable if and only if for every ε>0 there is a δ>0 such that for every initial condition x⃗0 for which ∥x⃗0∥<δ and for every time t>0 for which the solution x⃗(x⃗0,t) with the initial condition x⃗0 is defined, we have
(2.11)‖x⃗(x⃗0,t)‖<ε.
In the next theorem the well-known necessary conditions for the stability of the origin in x⃗′=x⃗*x⃗ are given.Theorem 2.10.
If an algebraA* contains an idempotent p⃗*p⃗=p⃗, then the origin in the corresponding system x⃗′=x⃗*x⃗ is unstable critical point.Proof.
First note thatIR·p⃗ is always a subalgebra of A*. Thus (by Proposition 2.5), the flow f(t)p⃗ is invariant. Since p⃗2=p⃗ from x⃗′=x⃗*x⃗ (when inserting x⃗(t)=f(t)p⃗) one obtains 1-dimensional ODE f′(t)=f2(t). Next observe that εp⃗ is in every neighborhood of the origin. Therefore the solution with the initial condition x⃗0=εp⃗ (i.e., f(0)=ε) is
(2.12)x⃗(t)=ε1-εt⋅p⃗;fort∈[0,1ε).
Finally observe that limt→1/ε∥x⃗(t)∥=∞ which completes the proof.Note that the immediate corollary of Theorem2.10 and the Kaplan-Yorke theorem is that systems x⃗′=x⃗*x⃗ with the stable origin always contain some nilpotents of rank two. In the continuous case the ray-solutions are as proven in Theorem 2.10 related 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 nilpotentn⃗ of rank two implies the existence of line of critical points IR·n⃗ in the continuous case, since from n⃗*n⃗=0⃗ one obtains (αn⃗)*(αn⃗)=α20⃗=0⃗ for every realα. However, in the discrete case the above property yields the existence of the ray-solution, since from x⃗k=αn⃗ one readily obtains that x⃗k+1=(αn⃗)*(αn⃗)=α20⃗=0⃗ for every real α.


## 3. Conclusions

For the stability analysis of the origin in systemsx⃗′=x⃗*x⃗ some new results are needed, for example, results obtained Markus approach in [13, 14]. Using Markus original classification one can obtain that only (up to linear equivalence) three (families of) systems admit stable origin in 2D. These systems are (cf. [13]):(3.1)x′=0,x′=-y2,x′=ky2,k<-1/8,y′=0,y′=2xy,y′=2xy+y2.In order to obtain similar result(s) inIR3 and/or in IRn for n>3 a partial algebraic classification (of systems/algebras with a plane of critical points) similar to Markus was done in (cf. [14]). Roughly speaking (cf. [13]), the existence of complex idempotents overlapping with the existence of the so-called essential nilpotents (i.e., nilpotents which are not contained in the linear span of all complex idempotents) seem to define (algebraically) the stability of the origin. The conjecture was confirmed by examining the complexification
(3.2)C*:=A*+iA*
of real algebras A* corresponding to the systems with a plane of critical points as well as on the so-called homogenized systems in IR3 (cf. [15]). It seems that [13] the spectral analysis of linear operator Ln⃗ defined by Ln⃗(a⃗):=a⃗*n⃗ (i.e., multiplication by essential nilpotent n⃗) is playing an important role in stability of the origin in systems x⃗′=x⃗*x⃗.However, algebraic approach is recently used (cf. [8, 9]) also in order to consider planar homogeneous discrete systems in the sense of (non)chaotic dynamics. The results are showing that the dynamics of systems whose corresponding algebras are containing some nilpotents of rank 2 cannot be chaotic [9]. Furthermore, system (1.3) is one of the simplest systems with chaotic dynamics and the corresponding algebra A(2) is power-associative. Note that every orbit of system x⃗k+1=x⃗k*x⃗k 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 x⃗0 the orbit of x⃗k+1=x⃗k*x⃗k can be obtained in terms of x⃗k+1=Lx⃗0(x⃗k), since in the power-associative algebras the powers of every x⃗0 are well defined (i.e., x⃗0n+1=x⃗0*x⃗0n=x⃗02*x⃗0n-1=⋯). In case of system (1.3) the left multiplication matrix of x⃗=(x,y) (by X⃗=(X,Y)) is obtained from x⃗*X⃗=(xX-yY,xY+yX)=[x-yyx][XY].In the chaotic region where∥x⃗∥=1, the corresponding multiplication matrix has the form
(3.3)L(cos⁡ϕ,sinϕ)=[cos⁡ϕ-sinϕsinϕcos⁡ϕ].
Readily, if ϕ=kπ where k is a rational number, then the point (cosϕ,sinϕ) is periodic. On the other hand, if ϕ=Kπ where K is irrational, the orbit of (cosϕ,sinϕ) is dense on the unit circle ∥x⃗∥=1 but not periodic. Furthermore, the points (cosϕ,sinϕ) (where ϕ=kπ and k is a rational number) are dense in ∥x⃗∥=1, as well. Thus there is chaos on ∥x⃗∥=1. 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 tox⃗′=x⃗*x⃗ with the initial condition x⃗(0)=x⃗0 can be expressed explicitly by the following formula:
(3.4)x⃗(t)=(I-tLx⃗0(x⃗0))-1(x⃗0),
where I is the identity matrix and Lx⃗0 is the linear operator defined by the (left) multiplication by x⃗0. The proof of the above explicit formula is a direct computation and can be found in [7], where (3.4) is used to prove that in power-associative algebras the corresponding system x⃗′=x⃗*x⃗ cannot have periodic solutions. Another interesting question (when considering power-associativity together with continuous quadratic systems) is whether one can use (3.4) in order to obtain some results on stability of the origin in IRn (n≥2).



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