On Algebraic Approach in Quadratic Systems

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.

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^′=𝑓(𝑥) or 𝑥_𝑘+1=𝑓(𝑥_𝑘) is defined to be the solution of the following algebraic (system of) equation(s), 𝑓(𝑥₀)=0 or 𝑓(𝑥₀)=𝑥₀, respectively. For the systems of ODEs, 𝑥^′=𝑓(𝑥), the critical point 𝑥₀ is said to be hyperbolic if no eigenvalue of the corresponding Jacobian matrix, 𝐽_𝑓(𝑥₀), of the (nonlinear vector) function 𝑓 has it is eigenvalue equal to zero (i.e., Re(𝜆_𝑖)≠0). In case of discrete system, 𝑥_𝑘+1=𝑓(𝑥_𝑘), the critical point 𝑥₀ is said to be hyperbolic if no eigenvalue of the Jacobian matrix has it is eigenvalue equal to 1 (i.e., |𝜆_𝑖|≠1). Roughly speaking, if for a continuous system Re(𝜆_𝑖)<0 for every 𝜆_𝑖, the corresponding critical point is stable (it is unstable, if Re(𝜆_𝑖)>0 for some 𝜆_𝑖). Similar, if for discrete systems |𝜆_𝑖|<1 for every 𝜆_𝑖, the corresponding critical point is stable (it is unstable, if |𝜆_𝑖|>1 for some 𝜆_𝑖). Note that just one eigenvalue of the corresponding linear approximation of 𝑥^′=𝑓(𝑥) or 𝑥_𝑘+1=𝑓(𝑥_𝑘) for which Re(𝜆_𝑖)=0 or |𝜆_𝑖|=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 form 𝑥^′=𝑄(𝑥) and autonomous homogeneous quadratic discrete dynamical systems of the form 𝑥_𝑘+1=𝑄(𝑥_𝑘) (where 𝑄∶IR^𝑛→IR^𝑛 is homogeneous of degree two in each component: 𝑄(𝑎𝑥)=𝑎²𝑄(𝑥) 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(𝑥,𝑦)=𝑥∗𝑦: 𝑄𝑦(𝑥)𝑦𝑥∗𝑦∶=𝑥+−𝑄−𝑄2 (1.1) in order to equip IR^𝑛 with a structure of a (nonassociative in general) commutative algebra (𝐴,∗). In the corresponding algebra (𝐴,∗) the square 𝑥∗𝑥=𝑥² of each vector 𝑥 is equal to 𝑥²=𝑄(2𝑥)−2𝑄(𝑥)2=2²𝑄(𝑥)−2𝑄(𝑥)2=𝑄(𝑥). (1.2) Thus, the system 𝑥^′=𝑄(𝑥) obviously becomes a Riccati equation 𝑥^′=𝑥∗𝑥=𝑥² 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 in IR² and in IR³ (the origin is namely a total degenerated critical point for 𝑥^′=𝑄(𝑥) in any dimension 𝑛 [3]).

However, in the discrete case 𝑥_𝑘+1=𝑄(𝑥_𝑘) 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 for𝑥_𝑘+1=𝑥²_𝑘−𝑦²_𝑘,𝑦_𝑘+1=2𝑥_𝑘𝑦_𝑘. (1.3)

Note that system (1.3) is a homogeneous quadratic (i.e., of the form 𝑥_𝑘+1=𝑄(𝑥_𝑘)) for𝑄(𝑥)=𝑄𝑥,𝑦=𝑥²−𝑦²,2𝑥𝑦. (1.4)

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:𝑥,𝑦∗(𝑢,𝑣)=𝑥𝑢−𝑦𝑣,𝑥𝑣+𝑦𝑢. (1.5)

Thus, in the standard basis 𝑒₁=(1,0) and 𝑒₂=(0,1) the multiplication table for the corresponding algebra is as illustrated in Table 1.

Table 1 

Applying 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 

On the other hand, beginning, for example, with the algebra 𝐀=(IR³,∗) given with the multiplication table as illustrated in Table 3 the corresponding quadratic form is obtained again by applying 𝑥∗𝑥=𝑥².

Table 3 

By denoting 𝑥=(𝑥,𝑦,𝑧)=𝑥𝑒₁+𝑦𝑒₂+𝑧𝑒₃, we get𝑥𝑒₁+𝑦𝑒₂+𝑧𝑒₃²=𝑥²𝑒²₁+𝑦²𝑒²₂+𝑧²𝑒²₃+2𝑥𝑦𝑒₁∗𝑒₂+2𝑥𝑧𝑒₁∗𝑒₃+2𝑦𝑧𝑒₂∗𝑒₃=𝑥²𝑒₁−𝑦²𝑒₁−𝑧²𝑒₁+2𝑥𝑦𝑒₂+2𝑥𝑧𝑒₃=𝑥²−𝑦²−𝑧².,2𝑥𝑦,2𝑥𝑧=𝑄𝑥,𝑦,𝑧 (1.6) Thus, we obtain the following quadratic systems 𝑥_𝑘+1=𝑥²_𝑘−𝑦²_𝑘−𝑧²_𝑘,𝑦_𝑘+1=2𝑥_𝑘𝑦_𝑘,𝑧_𝑘+1=2𝑥_𝑘𝑧_𝑘,𝑥^′=𝑥²−𝑦²−𝑧²,𝑦^′𝑧=2𝑥𝑦,_′=2𝑥𝑧. (1.7)

2. Some Connections between Systems and Their Corresponding Algebras

First note that the algebra which corresponds to a system 𝑥^′=𝑄(𝑥) or 𝑥_𝑘+1=𝑄(𝑥_𝑘) is always commutative, since from (1.1), it follows 𝑥∗𝑦=𝑦∗𝑥;∀𝑥,𝑦∈𝐴. (2.1) 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 0=𝑒₃∗𝑒₂∗𝑒₂≠𝑒₃∗𝑒₂∗𝑒₂=−𝑒₃. (2.2) 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 subset 𝑊⊆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 𝑥∈𝐴_∗ one can associate a subalgebra 𝑊_𝑥, defined by products 𝑥, 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 element 𝑥. 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 𝑥∈𝐴 we have 𝑥∗𝑖∈𝐼 and 𝑖∗𝑥∈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 map 𝜙∶𝐴→𝐵 is homomorphism from algebra (𝐴,∗) into algebra (𝐵,∘), if and only if, for every pair of vectors 𝑥,𝑦 from algebra 𝐴 we have: 𝜙(𝑥∗𝑦)=𝜙(𝑥)∘𝜙(𝑦). 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∶IR^𝑛→IR^𝑛 preserves solutions from system 𝑥^′=𝑥∗𝑥 into system 𝑦^′=𝑦∘𝑦 if and only if it takes parametrized solutions of the first system into parametrized solutions of the second one (i.e., 𝑦(𝑡)=ℎ(𝑥(𝑡)) is a solution of system 𝑆_∘, whenever 𝑥(𝑡) is a solution of 𝑆_∗. In discrete systems the solutions, 𝑥_𝑘; 𝑘=0,1,2,… are called orbits. By preserving of orbits we mean that ℎ(𝑥_𝑘); 𝑘=0,1,2,… is an orbit of system 𝑆_∘, whenever 𝑥_𝑘; 𝑘=0,1,2,… is an orbit of system 𝑆_∗.

Element 𝑎 of algebra (𝐴,∗) is said to be a nilpotent of rank 2, if 𝑎∗𝑎=0 and it is said to be an idempotent, if 𝑎∗𝑎=𝑎. If for some point 𝑥₀ the algebraic equation 𝑄(𝑥₀)=0 or 𝑄(𝑥₀)=𝑥₀ is fulfilled, it is called critical point of system 𝑥^′=𝑄(𝑥) or 𝑥_𝑘+1=𝑄(𝑥_𝑘), respectively. The solution 𝑥(𝑡) is a ray solution of 𝑥^′=𝑄(𝑥) if for every time 𝑡 vector 𝑥(𝑡) remains on the line IR𝑥(𝑡).

2.1. Algebraic Isomorphism and Linear Equivalence

The basic correspondence (1.1) between quadratic systems and algebras is the same for 𝑥^′=𝑄(𝑥) as well for 𝑥_𝑘+1=𝑄(𝑥_𝑘). The basic property concerning the linear equivalence between quadratic systems is also very similar as shown in the following two Propositions.

Proposition 2.1. Let 𝜙∶IR^𝑛→IR^𝑚 be linear. Then 𝜙 preserves solutions from system 𝑆_∗∶𝑥^′=𝑥∗𝑥; 𝑥∈IR^𝑛 into system 𝑆_∘∶𝑦^′=𝑦∗𝑦; 𝑦∈IR^𝑚 if and only if 𝜙 is a homomorphism from algebra 𝐴_∗=(IR^𝑛,∗) into algebra 𝐴_∘=(IR^𝑚,∘).

Proof. Let 𝜙 be some linear map which preserves solutions from 𝑆_∗ into 𝑆_∘. And let 𝐴_∗=(IR^𝑛,∗) and 𝐴_∘=(IR^𝑚,∘) be the corresponding algebras. Let 𝑥(𝑡) be the solution of 𝑆_∗ and let 𝑦(𝑡) be the solution of 𝑆_∘. Thus 𝑦=𝜙(𝑥) and 𝑥^′=𝑥∗𝑥 and from 𝑦^′=𝑦∘𝑦 one obtains (𝜙(𝑥))^′=𝜙(𝑥)∘𝜙(𝑥). Since 𝜙 is linear it is Jacobian is equal to 𝜙 in every point of the space (i.e., 𝜙^′=𝜙). Therefore 𝜙(𝑥)=𝜙(𝑥)∘𝜙(𝑥) for every 𝑥∈IR^𝑛. Substituting 𝑥=𝑋+𝑌 and applying commutativity and bilinearity of multiplications ∘ and ∗, we obtain 𝜙(𝑋∗𝑌)=𝜙(𝑋)∘𝜙(𝑌), for all 𝑋,𝑌∈IR^𝑛. Since 𝜙 is linear by assumption, this yields that 𝜙 is a homomorphism from 𝐴_∗=(IR^𝑛,∗) into 𝐴_∘=(IR^𝑚,∘).
Conversely, let 𝜙 be homomorphism from 𝐴_∗=(IR^𝑛,∗) into 𝐴_∘=(IR^𝑚,∘). Thus, for all 𝑋,𝑌∈IR^𝑛 we have 𝜙(𝑋∗𝑌)=𝜙(𝑋)∘𝜙(𝑌). For 𝑋=𝑌 we readily obtain: 𝜙(𝑋∗𝑋)=𝜙(𝑋)∘𝜙(𝑋). Using again 𝜙^′=𝜙, we obtain𝜙^′𝑋∗𝑋=𝜙𝑋∘𝜙𝑋. (2.3) Let 𝑋(𝑡) be a solution of 𝑆_∗. We want to prove that 𝜙(𝑋) is a solution of 𝑆_∘. Using 𝑋^′=𝑋∗𝑋 and (2.3) and the chain rule for the derivative one obtains 𝜙^′(𝑋^′)=𝜙^′(𝑋)⋅𝑋^′=(𝜙(𝑋))^′=𝜙(𝑋)∘𝜙(𝑋), which means that 𝜙(𝑋) is a solution of 𝑆_∘. This completes the proof.

Proposition 2.2. Let 𝜙∶IR^𝑛→IR^𝑚 be linear. Then 𝜙 preserves orbits from system 𝑆_∗∶𝑥_𝑘+1=𝑥_𝑘∗𝑥_𝑘; 𝑥∈IR^𝑛 into system 𝑆_∘∶𝑦_𝑘+1=𝑦_𝑘∗𝑦_𝑘; 𝑦∈IR^𝑚 if and only if 𝜙 is a homomorphism from algebra 𝐴_∗=(IR^𝑛,∗) into algebra 𝐴_∘=(IR^𝑚,∘).

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

The use of Propositions 2.1 and 2.2 is quite similar. In the following Example the use of Proposition 2.1 is considered.

Example 2.3. Systems 𝑆_(𝑥,𝑦)∶𝑥^′=𝑥²−𝑦²𝑦^′,𝑆=2𝑥𝑦_(𝑋,𝑌)∶𝑋^′=−62𝑋²−100𝑋𝑌−40𝑌²𝑌^′=85𝑋²+136𝑋𝑌+54𝑌² (2.4) are isomorphic. The corresponding isomorphism from (𝑥,𝑦) into (𝑋,𝑌) is −Φ=52−1232⁻¹. (2.5) 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 𝑋=−62𝑋²−100𝑋𝑌−40𝑌²𝑌=85𝑋²+136𝑋𝑌+54𝑌². (2.6) 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 to 𝑆(𝑋,𝑌).

Figure 2 

The particular solution to 𝑆(𝑥,𝑦).

Figure 3 

The phase diagram basis for 𝑆(𝑥,𝑦).

Figure 4 

The phase diagram basis for 𝑆(𝑋,𝑌).

The immediate corollary is that systems 𝑆_∗ 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 algebra 𝐴_∗=(IR^𝑚,∗) 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 system 𝑆_∗ has an invariant 𝑟-dimensional linear subspace 𝐸_𝑟 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., when 𝑆_∗∶𝑥_𝑘+1=𝑥_𝑘∗𝑥_𝑘; 𝑥∈IR^𝑛. The proof for continuous system 𝑆_∗∶𝑥^′=𝑥∗𝑥; 𝑥∈IR^𝑛 can be found, for example, in Markus [2].

Proof. Let 𝐸_𝑟=span(𝑒₁,𝑒₂,…,𝑒_𝑟) be an invariant r-dimensional linear subspace of a 𝑛-dimensional vector space 𝑉, 𝑛>𝑟. Then for every 𝑥∈𝐸_𝑟 the orbit {𝑥,𝑥²=𝑥∗𝑥,𝑥²∗𝑥,𝑥∗𝑥²,𝑥²∗𝑥²,(𝑥²∗𝑥)∗𝑥,(𝑥∗𝑥²)∗𝑥,…} is contained in 𝐸_𝑟. We will prove that (𝐸_𝑟,∗) is the 𝑟-dimensional subalgebra (i.e., the subspace 𝐸_𝑟 is closed for multiplication *). Setting 𝑥=𝑒_𝑖 we have 𝑒_𝑖∗𝑒_𝑖∈𝐸_𝑟 for every 1≤𝑖≤𝑟. Now we want to prove that 𝑒_𝑖∗𝑒_𝑗∈𝐸_𝑟 for all 1≤𝑖, 𝑗≤𝑟. In order to prove this, let us set 𝑥=𝑒_𝑖+𝑒_𝑗∈𝐸_𝑟 and compute 𝑥² using the commutativity rule in algebra 𝑥²=𝑒_𝑖+𝑒_𝑗∗𝑒_𝑖+𝑒_𝑗=𝑒_𝑖∗𝑒_𝑖+2𝑒_𝑖∗𝑒_𝑗+𝑒_𝑗∗𝑒_𝑗. (2.7) Since 𝑥²∈𝐸_𝑟, 𝑒_𝑖∗𝑒_𝑖∈𝐸_𝑟, and 𝑒_𝑗∗𝑒_𝑗∈𝐸_𝑟 it follows also 𝑒_𝑖∗𝑒_𝑗∈𝐸_𝑟 which means that for every pair 𝑥,𝑦∈𝐸_𝑟 the product 𝑥∗𝑦 is contained in 𝐸_𝑟, as stated. The converse follows directly from the fact that for every 𝑥,𝑦∈𝐸_𝑟, since 𝐸_𝑟 is a subalgebra, we have 𝑥∗𝑦∈𝐸_𝑟. Setting 𝑦=𝑥=𝑥₀ we immediately obtain that 𝑥₀∗𝑥₀∈𝐸_𝑟. Setting 𝑦=𝑥=𝑥₀∗𝑥₀ one obtains (𝑥₀∗𝑥₀)∗(𝑥₀∗𝑥₀)∈𝐸_𝑟, and so on. Thus the orbit {𝑥,𝑥²=𝑥∗𝑥,𝑥²∗𝑥,𝑥∗𝑥²,𝑥²∗𝑥²,(𝑥²∗𝑥)∗𝑥,(𝑥∗𝑥²)∗𝑥,…} is contained in 𝐸_𝑟, which means that 𝐸_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 to [10]).

Proposition 2.7. Let 𝐼 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 𝑥^′=𝑥∗𝑥 with the initial value problem 𝑥(0)=𝑥₀=𝑤₀+𝑖₀ can be solved by successive solution of 𝑤^′=𝑤∗𝑤;𝑤(0)=𝑤₀𝑖in𝑊,_′=𝑖∗𝑖+2𝑤(𝑡)∗𝑖;𝑖(0)=𝑖₀in𝐼, (2.8) where 𝑤(𝑡) is a solution of the first subsystem in 𝑊.

Corollary 2.8. A system 𝑥^′=𝑥∗𝑥 with the initial condition 𝑥(0)=𝑥₀ splits into two separated subsystems 𝑤^′=𝑤∗𝑤;𝑤(0)=𝑤₀𝑖in𝑊,_′=𝑖∗𝑖;𝑖(0)=𝑖₀in𝐼, (2.9) if and only if the corresponding algebra can be written as a direct sum of two nontrivial ideals 𝐴=𝐼₁⊕𝐼₂. (2.10)

Proof. Apply 𝐼=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 point 𝑥₀ of system 𝑥^′=𝑥∗𝑥 is said to be stable if and only if for every 𝜀>0 there is a 𝛿>0 such that for every initial condition 𝑥₀ for which ‖𝑥₀‖<𝛿 and for every time 𝑡>0 for which the solution 𝑥(𝑥₀,𝑡) with the initial condition 𝑥₀ is 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∗𝑥 are given.

Theorem 2.10. If an algebra 𝐴_∗ contains an idempotent 𝑝∗𝑝=𝑝, then the origin in the corresponding system 𝑥^′=𝑥∗𝑥 is unstable critical point.

Proof. First note that IR⋅𝑝 is always a subalgebra of 𝐴_∗. Thus (by Proposition 2.5), the flow 𝑓(𝑡)𝑝 is invariant. Since 𝑝²=𝑝 from 𝑥^′=𝑥∗𝑥 (when inserting 𝑥(𝑡)=𝑓(𝑡)𝑝) one obtains 1-dimensional ODE 𝑓^′(𝑡)=𝑓²(𝑡). Next observe that 𝜀𝑝 is in every neighborhood of the origin. Therefore the solution with the initial condition 𝑥₀=𝜀𝑝 (i.e., 𝑓(0)=𝜀) is 𝑥(𝑡)=𝜀⋅11−𝜀𝑡𝑝;for𝑡∈0,𝜀. (2.12) Finally observe that lim_{𝑡→1/𝜀}‖𝑥(𝑡)‖=∞ which completes the proof.

Note that the immediate corollary of Theorem 2.10 and the Kaplan-Yorke theorem is that systems 𝑥^′=𝑥∗𝑥 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 nilpotent 𝑛 of rank two implies the existence of line of critical points IR⋅𝑛 in the continuous case, since from 𝑛∗𝑛=0 one obtains (𝛼𝑛)∗(𝛼𝑛)=𝛼²0=0 for every real𝛼. However, in the discrete case the above property yields the existence of the ray-solution, since from 𝑥_𝑘=𝛼𝑛 one readily obtains that 𝑥_𝑘+1=(𝛼𝑛)∗(𝛼𝑛)=𝛼²0=0 for every real 𝛼.

3. Conclusions

For the stability analysis of the origin in systems 𝑥^′=𝑥∗𝑥 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]):𝑥^′=0,𝑥^′=−𝑦²,𝑥^′=𝑘𝑦²𝑦,𝑘<−1/8,_′=0,𝑦^′=2𝑥𝑦,𝑦^′=2𝑥𝑦+𝑦². (3.1)

In order to obtain similar result(s) in IR³ and/or in IR^𝑛 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 𝐶_∗∶=𝐴_∗+𝑖𝐴_∗ (3.2) 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 𝐿_𝑛 defined by 𝐿_𝑛(𝑎)∶=𝑎∗𝑛 (i.e., multiplication by essential nilpotent 𝑛) is playing an important role in stability of the origin in systems 𝑥^′=𝑥∗𝑥.

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 𝑥_𝑘+1=𝑥_𝑘∗𝑥_𝑘 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 𝑥_𝑘+1=𝑥_𝑘∗𝑥_𝑘 can be obtained in terms of 𝑥_𝑘+1=𝐿_𝑥0(𝑥_𝑘), since in the power-associative algebras the powers of every 𝑥₀ are well defined (i.e., 𝑥₀^𝑛+1=𝑥₀∗𝑥^𝑛₀=𝑥²₀∗𝑥₀^𝑛−1=⋯). In case of system (1.3) the left multiplication matrix of 𝑥=(𝑥,𝑦) (by 𝑋=(𝑋,𝑌)) is obtained from 𝑥∗𝑋=(𝑥𝑋−𝑦𝑌,𝑥𝑌+𝑦𝑋)=^𝑥−𝑦_𝑦𝑥^𝑋_𝑌.

In the chaotic region where ‖𝑥‖=1, the corresponding multiplication matrix has the form 𝐿_{(cos𝜙,sin𝜙)}=cos𝜙−sin𝜙sin𝜙cos𝜙. (3.3) 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 ‖𝑥‖=1 but not periodic. Furthermore, the points (cos𝜙,sin𝜙) (where 𝜙=𝑘𝜋 and 𝑘 is a rational number) are dense in ‖𝑥‖=1, as well. Thus there is chaos on ‖𝑥‖=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 to 𝑥^′=𝑥∗𝑥 with the initial condition 𝑥(0)=𝑥₀ can be expressed explicitly by the following formula: 𝑥(𝑡)=𝐼−𝑡𝐿_𝑥0𝑥₀⁻¹𝑥₀, (3.4) where 𝐼 is the identity matrix and 𝐿_𝑥0 is the linear operator defined by the (left) multiplication by 𝑥₀. 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 𝑥^′=𝑥∗𝑥 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 IR^𝑛 (𝑛≥2).