Creation of single-wing Lorenz-like attractors via a ten-ninths-degree term

Jun Pan; Haijun Wang; Guiyao Ke; Feiyu Hu

doi:10.1515/phys-2025-0165

Article Open Access

Creation of single-wing Lorenz-like attractors via a ten-ninths-degree term

Jun Pan , Haijun Wang , Guiyao Ke and Feiyu Hu

Published/Copyright: June 5, 2025

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From the journal Open Physics Volume 23 Issue 1

Abstract

In light of the subtle connection between the strange attractors and the degree of dynamical systems, in this study, we propose a new simple asymmetric Lorenz-like system and report the finding of single-wing Lorenz-like attractors, which can also be created by the collapse of asymmetric singularly degenerate heteroclinic cycles. Moreover, we prove the existence of a single heteroclinic orbit to the origin and the nontrivial equilibrium point. Besides, numerical simulations also verify the theoretical analysis.

Keywords: heteroclinic orbit; generalization of the second part of Hilbert’s 16th problem; Lyapunov function; single-wing attractor; asymmetric singularly degenerate heteroclinic cycle

1 Introduction

As early as 1990, Cox introduced an asymmetric perturbation of the Lorenz system and illustrated the transition to complicated dynamics in parameter-space, i.e., the classic Lorenz system with a constant ε in its second equation, to meet the need of the scenario of the real-life laboratory set-up [1]. Enlightened by this thought, Lü et al. proposed the controlled Lü and Chen system and performed a mirror operation to merge together two simple single-wing attractors to obtain the compound structure of the Lü attractor and Chen attractor [2,3]. Unfortunately, Liu et al. revisited the controlled Chen system and discovered the significant difference between the two aforementioned simple single-wing attractors and the Chen attractor, i.e., Lyapunov exponents and Lyapunov dimensions [4]. Meanwhile, Miranda and Stone transformed the original Lorenz system into the proto-Lorenz system, interpreting the Lorenz system as a double covering of the proto-Lorenz system [5].

On the one hand, the controlled Lorenz, Chen, and Lü systems are potential candidates that can generate single-wing Lorenz-like attractors, due to the cause of a constant controller. On the other hand, if the constant controller ε is considered as the variables with power of zero, i.e., ε x 0 , or ε y 0 , or ε z 0 , etc., then the controlled Lorenz, Chen, and Lü systems also enrich the generalization of Hilbert’s 16th problem [6–8]. This implies that the degree of polynomials in the model under study may have something to do with the geometric structure of attractors and repellers (as long as they exist), and not only to their number and mutual position.

Researchers have extensively investigated the relationship between polynomial degrees and complex dynamics, particularly in systems exhibiting Lorenz-like attractors and singular orbits. In 2007, replacing x 1 of the Lorenz system with ϕ 1 ( x 1 ) = a x 1 ( x 1 − 1 ) ( x 1 − 2 ) and coining two two-scroll Lorenz attractors starting from the initial conditions (2.524, 1, 1) and (2.526, 1, 1), Chen and Zhang proposed an open problem: degrees of ϕ i ( x i ) control the number of the attractors of x ˙ = Φ ( ϕ 1 ( x 1 ) , ϕ 2 ( x 2 ) , ϕ 3 ( x 3 ) ) , i = 1, 2, 3 [9]. In 2022, Wang et al. introduced x 3 into the Yang system’s second and third equations, revealing many hidden attractors [10,11]. Later on, by introducing x 5 , they performed the same operation and illustrated more Lorenz-like attractors [12], and demonstrated that similar substitutions could expand the parameter ranges for chaotic attractors (they may be transient chaos, self-excited or hidden ones) that coexist with the unstable origin and a pair of stable node-foci. In 2024, Ke et al. [13] changed x 3 back to x for the Lorenz-like system [11] and found more self-excited Lorenz-like attractors in contrast to the sub-quadratic one [10]. By replacing x of the extended Lorenz system with sin ( x ) , Wang et al. found conjoined Lorenz-like attractors [14]. By inserting x 2 to the Yang system’s second equation, Wang et al. revealed asymmetric singularly degenerate heterocliniccycles (SDHC) with nearby asymmetric two-wing attractors [15]. However, substituting x 2 for x , Li and Wang obtained the modified cubic Lorenz-like system and did not observe SDHC and chaotic attractors [16]. By changing x to x 2 3 , the new derived sub-quadratic Lorenz-like system has neither SDHC nor strange attractors [17]. Fortunately, the aforementioned Lorenz-like systems were proved to have a single/pair of heteroclinic orbits using different Lyapunov functions.

However, now the question is: how to create single-wing Lorenz-like attractors through changing the degrees of state variables, if they exist? To the best of our knowledge, scholars seldom consider this topic. More importantly, studying such a system may shed light on the research hotspot, i.e., data-driven predictions of the chaotic system [18–21]. To achieve this target, by increasing the degree of the linear term c x of the Yang system to 10 9 , we achieve the following contributions:

Creating single-wing Lorenz-like attractors by a ten-ninths-degree term.
Explaining the formation of single-wing Lorenz-like attractors through collapses of asymmetric SDHC.
Establishing the existence of a heteroclinic orbit connecting S 0 and S ′ .

This proposed quadratic Lorenz-like system aligns with the third principle of Sprott [22], representing the Lorenz-like analogue with only six terms. It addresses a critical gap in traditional Lorenz-like models by leveraging fractional-degree nonlinearity to generate asymmetric dynamics. Unlike controlled Lorenz/Chen systems, which rely on constant perturbations, our model uses a ten-ninths-degree term x 10 9 to create single-wing attractors, offering a novel paradigm for studying how algebraic structures shape attractor geometry, contributing to the generalization of Hilbert’s 16th problem, where polynomial degree is shown to govern not only the number of attractors but also their topological complexities. Furthermore, the results obtained may provide insights into the fractional analogues that generate single-wing Lorenz-like attractors [23], the numerical algorithm [24], and model-free prediction of chaotic dynamics [25], etc.

The remainder of this work is structured as follows. Section 2 details the construction of the new simple asymmetric Lorenz-like system and reports the main results. Section 3 rigorously demonstrates the presence of a single heteroclinic orbit to S 0 and S ′ when 0 < 2 a ≤ b and c ≠ 0 . At last, Section 4 concludes the study and explores prospective research directions, with a focus on investigating hidden single-wing Lorenz-like attractors.

2 New Lorenz-like system and key results

By comparing those Lorenz-like systems [10– 13,15–17] and systematic experimentation, this section first introduces the following new asymmetric Lorenz-like system:

(2.1) x ˙ = a ( y − x ) , y ˙ = c x 10 9 − x z , z ˙ = − b z + x y ,

where a ≠ 0 , b , c ∈ R , and summarizes the local dynamics in the following theorem.

Theorem 2.1

Table 1 presents the equilibria of system (2.1).
Tables 2 and 3 list the local behaviors of S 0 and S z , where W loc s / W loc c / W loc u denote the locally stable/center/unstable manifolds, as shown in the study by Kuzenetsov [26].
Table 4 lists the local behaviors of S ′ , where W = { ( a , b , c ) ∣ a b c ≠ 0 } , W 1 = { ( a , b , c ) ∈ W ∣ a + b > 0 , a b + ( 1 − a 9 b ) ( b c ) 18 17 > 0 , 17 a ( b c ) 18 17 9 > 0 , W 2 = W \ W 1 , △ = a b ( a + b ) − ( 9 a b − 9 b 2 + a 2 ) ( b c ) 18 17 9 b , c * = 1 b ( 9 a b 2 ( a + b ) 9 a b − 9 b 2 + a 2 ) 17 18 , W 1 3 = { ( a , b , c ) ∈ W 1 ∣ △ > 0 } , W 1 2 = { ( a , b , c ) ∈ W 1 ∣ c = ± c * } , and W 1 1 = { ( a , b , c ) ∈ W 1 ∣ △ < 0 } .

Table 1

Equilibrium points

b	c	Distribution of equilibria
= 0		S z = { ( 0,0 , z ) ∣ z ∈ R }
≠ 0	= 0	S 0 = ( 0 , 0 , 0 )
	≠ 0	S 0 = ( 0,0,0 ) , S ′ = ( ( b c ) 9 17 , ( b c ) 9 17 , ( b c ) 18 17 b )

Table 2

Local dynamics of S 0

b	a	c	Property of S 0
< 0	< 0		A 3D W loc u
	> 0		A 1D W loc s and a 2D W loc u
> 0	< 0	= 0	A 2D W loc s and a 1D W loc u
		≠ 0	A 1D W loc s and a 2D W loc u
	> 0	= 0	A 3D W loc s
		≠ 0	A 2D W loc s and a 1D W loc u

Table 3

Local dynamics of S z

a	z	Property of S z
< 0	< 0	A 1D W loc c and a 2D W loc u
	> 0	A 1D W loc s , a 1D W loc c and a 1D W loc u
> 0	< 0	A 1D W loc s , a 1D W loc c and a 1D W loc u
	> 0	A 2D W loc s and a 1D W loc c

Table 4

Local dynamics of S ′

( a , c , b )	Property of S ′
W 1 1 ∪ W 2	Unstable
W 1 2	Hopf bifurcation
W 1 3	Asymptotically stable

Linear stability analysis for S 0 and S z yields straightforward results, which are omitted here.

For S ′ , the characteristic equation of the Jacobian matrix is calculated as

(2.2) λ 3 + ( a + b ) λ 2 + a b + ( 1 − a 9 b ) ( b c ) 18 17 λ + 17 a ( b c ) 18 17 9 = 0 ,

from which, the local stability of S ′ is determined by the Routh–Hurwitz criterion. For ( a , b , c ) ∈ W 1 2 , λ 1 = − ( a + b ) and λ 2,3 = ± ω i = ± 17 a 2 b 2 9 a b − 9 b 2 + a 2 i are eigenvalues of S ′ , i.e., the roots of Eq. (2.2). Further, taking derivative on Eq. (2.2) with respect to c and replacing λ with ω i , one obtains d Re ( λ 2 ) d c ∣ c = ± c * = ± b c * 17 ( 9 a b − 9 b 2 + a 2 ) 17 [ ω 2 + ( a + b ) 2 ] ≠ 0 to verify the transversal condition, which leads to Hopf bifurcation at S ′ , as shown in Figure 1. For the stability of Hopf bifurcation of S ′ , the interested readers can apply the project method to determine it [26,27].

$Figure 1 When ( a , b ) \left(a,b) = (5, 1), (a) c = − 4.0751 c=-4.0751 , ( x 0 1 , y 0 1 , z 0 1 ) = ( − 2.323 , − 2.408 , 4.429 ) \left({x}_{0}^{1},{y}_{0}^{1},{z}_{0}^{1})=\left(-2.323,-2.408,4.429) , (b) c = 4.0751 c=4.0751 , ( x 0 2 , y 0 2 , z 0 1 ) \left({x}_{0}^{2},{y}_{0}^{2},{z}_{0}^{1}) = (2.323, 2.408, 4.429), a closed orbit around S ′ {S}^{^{\prime} } when undergoing Hopf bifurcation. (a) ( a , c , b ) = ( 5 , − 4.0751 , 1 ) \left(a,c,b)=\left(5,-4.0751,1) and (b) ( a , c , b ) \left(a,c,b) = (5, 4.0751, 1).$

Figure 1

When ( a , b ) = (5, 1), (a) c = − 4.0751 , ( x 0 1 , y 0 1 , z 0 1 ) = ( − 2.323 , − 2.408 , 4.429 ) , (b) c = 4.0751 , ( x 0 2 , y 0 2 , z 0 1 ) = (2.323, 2.408, 4.429), a closed orbit around S ′ when undergoing Hopf bifurcation. (a) ( a , c , b ) = ( 5 , − 4.0751 , 1 ) and (b) ( a , c , b ) = (5, 4.0751, 1).

Second, based on Theorem 2.1 and using Matlab’s procedure ode45 and Wolf Lyapunov exponent estimation [28], we delve more deeply into system (2.1) in the following single-wing Lorenz-like attractors and asymmetric SDHC.

For parameters ( a , b ) = ( 5 , 1 ) and ( x 0 3 , y 0 3 , z 0 2 ) = ( 1.314 , 1.618 , 2.236 ) , Figure 2 shows the bifurcation diagrams and Lyapunov exponents vs the parameter c ∈ [ − 100 , 100 ] . Interestingly, the chaotic behaviors illustrated in Figure 1 display only single-wing Lorenz-like attractors rather than the two-wing ones, as shown in Figure 3.

$Figure 2 When ( a , b ) = ( 5 , 1 ) \left(a,b)=\left(5,1) and ( x 0 3 , y 0 3 , z 0 2 ) \left({x}_{0}^{3},{y}_{0}^{3},{z}_{0}^{2}) = (1.314, 1.618, 2.236), (a), (b), (c) bifurcation diagrams and (d) Lyapunov exponents ( λ i , i = 1 , 2 , 3 {\lambda }_{i},i=1,2,3 ) vs the parameter c c , yielding that system (2.1) experiences stable equilibrium point, Hopf bifurcation, period attractors, chaotic attractors when the parameter c c varies. (a) c − x c-x , (b) c − y c-y , (c) c − z c-z , and (d) Lyapunov exponents vs the parameter c c .$

Figure 2

When ( a , b ) = ( 5 , 1 ) and ( x 0 3 , y 0 3 , z 0 2 ) = (1.314, 1.618, 2.236), (a), (b), (c) bifurcation diagrams and (d) Lyapunov exponents ( λ i , i = 1 , 2 , 3 ) vs the parameter c , yielding that system (2.1) experiences stable equilibrium point, Hopf bifurcation, period attractors, chaotic attractors when the parameter c varies. (a) c − x , (b) c − y , (c) c − z , and (d) Lyapunov exponents vs the parameter c .

$Figure 3 When ( a , b ) = ( 5 , 1 ) \left(a,b)=\left(5,1) and ( x 0 3 , y 0 3 , z 0 2 ) = ( 1.314 , 1.618 , 2.236 ) \left({x}_{0}^{3},{y}_{0}^{3},{z}_{0}^{2})=\left(1.314,1.618,2.236) , (a) c = 18 c=18 , (b) c = − 18 c=-18 , single-wing Lorenz-like attractors depicted in Figure 2. (a) c = 18 c=18 and (b) c = − 18 c=-18 .$

Figure 3

When ( a , b ) = ( 5 , 1 ) and ( x 0 3 , y 0 3 , z 0 2 ) = ( 1.314 , 1.618 , 2.236 ) , (a) c = 18 , (b) c = − 18 , single-wing Lorenz-like attractors depicted in Figure 2. (a) c = 18 and (b) c = − 18 .

Third, set ( a , c ) = ( 0.5 , ± 6 ) and ( x 0 4,5 , y 0 4,5 ) = ( ± 1.314 , ± 1.618 ) × 1 0 − 7 , z 0 3,4,5 = − 1 , − 3 , − 5 , and b ∈ [ 0 , 1 ] . In order to detect asymmetric SDHC and their proximity to single-wing Lorenz-like attractors, we resort to the bifurcation diagrams and Lyapunov exponents vs the parameter c , as illustrated in Figures 4 and 5. Moreover, based on the dynamics of S z listed in Table 3, we make a numerical study below.

$Figure 4 When ( a , c ) = ( 0.5 , 6 ) \left(a,c)=\left(0.5,6) and ( x 0 4 , y 0 4 , z 0 3 ) \left({x}_{0}^{4},{y}_{0}^{4},{z}_{0}^{3}) = ( 1.314 × 1 0 − 7 , 1.618 × 1 0 − 7 , − 1 ) \left(1.314\times 1{0}^{-7},1.618\times 1{0}^{-7},-1) , (a), (b), (c) bifurcation diagrams and (d) Lyapunov exponents vs the parameter b b , yielding the existence of SDHC with nearby bifurcated single-wing Lorenz-like attractors when the parameter b > 0 b\gt 0 is slightly perturbed. (a) b − x b-x , (b) b − y b-y , (c) b − z b-z , (d) Lyapunov exponents vs the parameter b b .$

Figure 4

When ( a , c ) = ( 0.5 , 6 ) and ( x 0 4 , y 0 4 , z 0 3 ) = ( 1.314 × 1 0 − 7 , 1.618 × 1 0 − 7 , − 1 ) , (a), (b), (c) bifurcation diagrams and (d) Lyapunov exponents vs the parameter b , yielding the existence of SDHC with nearby bifurcated single-wing Lorenz-like attractors when the parameter b > 0 is slightly perturbed. (a) b − x , (b) b − y , (c) b − z , (d) Lyapunov exponents vs the parameter b .

$Figure 5 When ( a , c ) = ( 0.5 , − 6 ) \left(a,c)=\left(0.5,-6) and ( x 0 5 , y 0 5 , z 0 3 ) \left({x}_{0}^{5},{y}_{0}^{5},{z}_{0}^{3}) = ( − 1.314 × 1 0 − 7 , − 1.618 × 1 0 − 7 , − 1 ) \left(-1.314\times 1{0}^{-7},-1.618\times 1{0}^{-7},-1) , (a), (b), (c) bifurcation diagrams and (d) Lyapunov exponents vs the parameter b b , yielding the existence of SDHC with nearby bifurcated single-wing Lorenz-like attractors when the parameter b > 0 b\gt 0 is slightly perturbed. (a) b − x b-x , (b) b − y b-y , (c) b − z b-z , (d) Lyapunov exponents vs the parameter c c .$

Figure 5

When ( a , c ) = ( 0.5 , − 6 ) and ( x 0 5 , y 0 5 , z 0 3 ) = ( − 1.314 × 1 0 − 7 , − 1.618 × 1 0 − 7 , − 1 ) , (a), (b), (c) bifurcation diagrams and (d) Lyapunov exponents vs the parameter b , yielding the existence of SDHC with nearby bifurcated single-wing Lorenz-like attractors when the parameter b > 0 is slightly perturbed. (a) b − x , (b) b − y , (c) b − z , (d) Lyapunov exponents vs the parameter c .

Numerical Study 2.1 Set c = 6 (resp. − 6 ), a = 0.5 , b = 0 , and ( x 0 4 , y 0 4 ) = ( 1.314 , 1.618 ) × 1 0 − 7 (resp. ( x 0 5 , y 0 5 ) = − ( 1.314 , 1.618 ) × 1 0 − 7 ), z 0 3,4,5 = − 1 , − 3 , − 5 , Figure 6(a); resp. Figure 6(c) the one-dimensional unstable manifold W u ( S z 1,2,3 ) ( S z 1,2,3 = ( 0 , 0 , − 1 ) , ( 0 , 0 , − 3 ) , ( 0 , 0 , − 5 ) ) tends to the stable (0, 0, 11.9519), (0, 0, 13.4969), and (0, 0, 15.0369) when t → ∞ , generating asymmetric SDHC. When the parameter b > 0 is slightly perturbed, the collapse of these heteroclinic cycles creates a single-wing Lorenz-type attractor, as shown in Figure 6(b) (resp. Figure 6(d)).

Figure 6

For a = 0.5 , (a) b = 0 , (b) b = 0.06 , ( x 0 4 , y 0 4 ) = ( 1.314 , 1.618 ) × 1 0 − 7 , z 0 3 , 4 , 5 = − 1 , − 3 , − 5 , (color in red, blue, and yellow), (c) b = 0 , (d) b = 0.06 , ( x 0 5 , y 0 5 ) = − ( 1.314 , 1.618 ) × 1 0 − 7 , z 0 3 , 4 , 5 = − 1 , − 3 , − 5 (color in magenta, green, and cyan), asymmetric SDHC with nearby single-wing Lorenz-like attractors suggested in Figures 4 and 5. (a) SDHC for ( a , c , b ) = ( 0.5 , 6 , 0 ) , (b) single-wing Lorenz-like attractor for ( a , c , b ) = ( 0.5 , 6 , 0.06 ) , (c) SDHC for ( a , c , b ) = ( 0.5 , − 6 , 0 ) , and (d) single-wing Lorenz-like attractor for ( a , c , b ) = ( 0.5 , − 6 , 0.06 ) .

Finally, using two appropriate Lyapunov functions and imitating the ones from previous literature [10–17, 29–38], we prove the following theorem in Section 3.

Theorem 2.2

If b ≥ 2 a > 0 , c ≠ 0 , then

closed orbits are nonexistent in system (2.1);
system (2.1) does not have any homoclinic orbits;
a single heteroclinic orbit to S 0 and S ′ only exists in system (2.1).

In order to enhance the readability of the proof of the Section 3, the following symbols are introduced:

ψ t ( p 0 ) = ( x ( t ; x 0 ) , y ( t ; y 0 ) , z ( t ; z 0 ) ) : every one orbit of system (2.1) originating from the initial condition p 0 = ( x 0 , y 0 , z 0 ) .
W ∓ u : the negative and positive branches of the unstable manifold W u ( S 0 ) corresponding to x > 0 and x < 0 as t → − ∞ .

3 Heteroclinic orbit

In this section, we first give the following two Lyapunov functions and the corresponding derivatives along ψ t ( p 0 ) (1) b − 2 a > 0

V 1 ( ψ t ( p 0 ) ) = b ( b − 2 a ) ( y − x ) 2 + ( − b z + x 2 ) 2 + 9 ( b − 2 a ) 19 a ( − b c x 9 + x 2 ) 2 + ( b − 2 a ) 38 a ( − ( b c ) 18 17 + x 2 ) 2 + 8 ( b − 2 a ) ( b c ) 32 17 19 a ( − ( b c ) 2 17 + x 2 9 ) 2 + 4 ( b − 2 a ) ( b c ) 30 17 19 a ( − ( b c ) 2 17 x 9 + x 3 9 ) 2 + 2 ( b − 2 a ) ( b c ) 26 17 19 a ( − ( b c ) 4 17 x 9 + x 5 9 ) 2 + ( b − 2 a ) ( b c ) 18 17 19 a ( − ( b c ) 8 17 x 9 + x ) 2 ,

(3.1) d V 1 ( ψ t ( p 0 ) ) d t ( 2.1 ) = 2 a b ( 2 a − b ) ( x − y ) 2 − 2 b ( x 2 − b z ) 2 .

(2) 2 a = b > 0

V 2 ( ψ t ( p 0 ) ) = 38 ( y − x ) 2 + 18 ( − 2 a c x 9 + x 2 ) 2 + 16 ( 2 a c ) 32 17 ( − ( 2 a c ) 2 17 + x 2 9 ) 2 + 8 ( 2 a c ) 30 17 ( − ( 2 a c ) 2 17 x 9 + x 3 9 ) 2 + 4 ( 2 a c ) 26 17 ( − ( 2 a c ) 4 17 x 9 + x 5 9 ) 2 + 2 ( 2 a c ) 18 17 ( − ( b c ) 8 17 x 9 + x ) 2 + ( − ( 2 a c ) 18 17 + x 2 ) 2 ,

(3.2) d V 2 ( ψ t ( p 0 ) ) d t ( 2.1 ) = − 76 a ( x − y ) 2 .

Following the procedure in [10–17,29–39], one has to prove Lemma 3.1.

Lemma 3.1

Suppose c ≠ 0 and 0 < 2 a ≤ b . The following statements hold.

If ∃ t i , i = 1 , 2 , t 2 > t 1 , and V 1,2 ( ψ t 1 ( p 0 ) ) = V 1 , 2 ( ψ t 2 ( p 0 ) ) , then p 0 ∈ { S 0 , S ′ } .
If ∃ t ∈ R , x ( t ; x 0 ) > 0 (resp. < 0 ), and ψ t ( p 0 ) → S 0 as t → − ∞ , then ∀ t ∈ R , V 1,2 ( S 0 ) > V 1,2 ( ψ t ( p 0 ) ) , and x ( t ; x 0 ) > 0 (resp. < 0 ). Namely, p 0 ∈ W + u (resp. W − u ).

Proof

(i) Based on the hypothesis of (i) and Eqs (3.1) and (3.2), the fact d V 1,2 ( ψ t ( p 0 ) ) d t = 0 for all t ∈ ( t 1 , t 2 ) yields

(3.3) x ˙ ( t ; x 0 ) ≡ z ˙ ( t ; z 0 ) ≡ y ˙ ( t ; y 0 ) ≡ 0 ,

i.e., p 0 ∈ { S 0 , S ′ } . Precisely speaking, if b = 2 a > 0 , the hypothesis of (i) suggests x 2 ≡ 2 a z . In fact, Q ( ψ t ( p 0 ) ) = x 2 − 2 a z with d Q ( ψ t ( p 0 ) ) d t ∣ ( 2.1 ) = − 2 a Q ( ψ t ( p 0 ) ) leads to

(3.4) Q ( ψ t ( p 0 ) ) = Q ( ψ τ ( p 0 ) ) e − 2 a ( t − τ ) for all τ , t ∈ R .

Since the boundedness of ψ τ ( p 0 ) as τ → − ∞ , Eq. (3.4) yields Q ( ψ t ( p 0 ) ) ≡ 0 , i.e., x 2 ≡ 2 a z . When 0 < 2 a < b , we arrive at ψ t ( p 0 ) ∈ { x 2 − b z = 0 } ∩ { y − x = 0 } leads to z ˙ ( t , z 0 ) ≡ x ˙ ( t , x 0 ) ≡ 0 , resulting in x ( t ) = y ( t ) = x 0 and thus, y ˙ ( t , y 0 ) = 0 , ∀ t ∈ R .

(ii) First, we prove V 1,2 ( S 0 ) > V 1,2 ( ψ t ( p 0 ) ) , ∀ t ∈ R . Otherwise, ∃ t 0 ∈ R , 0 < V 1,2 ( S 0 ) ≤ V 1,2 ( ψ t 0 ( p 0 ) ) holds. Based on the continuity of V 1,2 in t and ψ t ( p 0 ) → S 0 as t → − ∞ , ∃ t n → − ∞ and n 1,2 > 0 , ∣ V 1,2 ( ψ t n ( p 0 ) ) − V 1,2 ( S 0 ) ∣ < ε 1,2 is true, ∀ ε 1,2 > 0 and n > max { n 1 , n 2 } . Due to t n → − ∞ and t 0 ∈ R , ∃ n 3,4 , t 0 > t n holds, for all max { n 3,4 } < n . Set n 0 = max { n 1,2,3,4 } and ε 1,2 = 1 2 [ V 1,2 ( ψ t 0 ( p 0 ) ) − V 1,2 ( S 0 ) ] ≥ 0 . Then, V 1,2 ( ψ t n ( p 0 ) ) − V 1,2 ( ψ t 0 ( p 0 ) ) = V 1,2 ( ψ t n ( p 0 ) ) − V 1,2 ( S 0 ) + V 1,2 ( S 0 ) − V 1,2 ( ψ t 0 ( p 0 ) ) < ε 1,2 + V 1,2 ( S 0 ) − V 1,2 ( ψ t 0 ( p 0 ) ) = − ε 1,2 ≤ 0 . Instead, the fact d V 1,2 ( ψ t ( p 0 ) ) d t ≤ 0 in Eqs. (3.1) and (3.2) results in V 1,2 ( ψ t n ( p 0 ) ) ≥ V 1,2 ( ψ t 0 ( p 0 ) ) , ∀ t n < t 0 , n > n 0 . As a result, V 1,2 ( ψ t n ( p 0 ) ) = V 1,2 ( ψ t 0 ( p 0 ) ) and the hypothesis of (i) leads to p 0 ∈ { S 0 , S ′ } . Because ψ t ( p 0 ) → S 0 , p 0 ≡ S 0 , and x ( t , x 0 ) ≡ 0 hold, ∀ t ∈ R , which yet conflict with the assumed condition that x ( t , x 0 ) > 0 (resp. x ( t , x 0 ) < 0 ), ∃ t ∈ R . Therefore, V 1,2 ( S 0 ) > V 1,2 ( ψ t ( p 0 ) ) , ∀ t ∈ R .

Further, we prove that x ( t , x 0 ) > 0 (resp. x ( t , x 0 ) < 0 ), for all t ∈ R . Otherwise, there exists t ′ ∈ R such that x ( t ′ , x 0 ) ≤ 0 (resp. x ( t ′ , x 0 ) ≥ 0 ) holds. On the other hand, ∃ t ″ ∈ R , it follows from the hypothesis of (ii) that x ( t ″ , x 0 ) > 0 (resp. x ( t ″ , x 0 ) < 0 ) is also true. Therefore, ∃ τ ∈ R , x ( τ , x 0 ) = 0 holds. Due to V 1,2 ( ψ t ( p 0 ) ) < V 1,2 ( S 0 ) , ∀ t ∈ R , it follows that ψ τ ( p 0 ) ∈ { ψ t ( p 0 ) ∣ V 1 ( ψ t ( p 0 ) ) < V 1 ( S 0 ) } ∩ { ψ t ( p 0 ) ∣ x = 0 } = { ψ t ( p 0 ) ∣ b ( b − 2 a ) y 2 + ( b z ) 2 + 17 ( b − 2 a ) ( b c ) 36 17 38 a < 17 ( b − 2 a ) ( b c ) 36 17 38 a } = ∅ for V 1 , and ψ τ ( p 0 ) ∈ { ψ t ( p 0 ) ∣ 38 x 2 + 17 ( 2 a c ) 36 17 < 17 ( 2 a c ) 36 17 } = ∅ for V 2 , leading to a contradiction. As a result, x ( t , x 0 ) > 0 (resp. x ( t , x 0 ) < 0 ), ∀ t ∈ R . This completes the proof.□

Next the proof of Theorem 2.2 easily follows from Lemma 3.1.

Proof

(a) As d V 1,2 ( ψ t ( p 0 ) ) d t ≤ 0 in Eqs (3.1) and (3.2) for b ≥ 2 a > 0 , c ≠ 0 , we obtain

(3.5) 0 ≤ V 1,2 ( ψ t ( p 0 ) ) ≤ V 1,2 ( p 0 ) ,

which suggests that lim t → + ∞ V 1,2 ( ψ t ( p 0 ) ) = V 1,2 * ( p 0 ) exists. Namely, the boundedness of V 1,2 ( ψ t ( p 0 ) ) also suggests that ψ t ( p 0 ) is bounded for all t ≥ 0 . Let Ω ( p 0 ) denote the ω -limit set of the orbit ψ t ( p 0 ) . In a word, if q ∈ Ω ( p 0 ) , then every orbit through q belongs to Ω ( p 0 ) , i.e., ψ t ( p 0 ) ∈ Ω ( p 0 ) . Consequently, ∀ ψ t ( q ) , t ≥ 0 , ∃ t n → ∞ for n → ∞ , lim n → + ∞ ψ t n ( p 0 ) = ψ t ( q ) , leading to

V 1,2 ( ψ t ( q ) ) = lim n → + ∞ V 1,2 ( ψ t n ( p 0 ) ) = V 1,2 * ( p 0 ) = const ,

∀ t ≥ 0 . Based on Lemma 3.1, q ∈ { S 0 , S ′ } .

(b) Assume γ ( t , p 0 ) is a homoclinic orbit to S 0 or S ′ , i.e., lim t → ± ∞ γ ( t , p 0 ) = o , where p 0 ∉ { S 0 , S ′ } and o ∈ { S 0 , S ′ } . The fact d V 1,2 ( ψ t ( p 0 ) ) d t ≤ 0 in Eqs (3.1) and (3.2) leads to

0 ≤ V 1,2 ( o ) = V 1,2 ( γ ( − ∞ , p 0 ) ) ≤ V 1,2 ( γ ( t , p 0 ) ) ≤ V 1,2 ( γ ( ∞ , p 0 ) ) = V 1,2 ( o ) ,

i.e. V 1,2 ( γ ( t , p 0 ) ) = V 1,2 ( o ) , ∀ t ∈ R . According to Lemma 3.1(i), it follows that p 0 ∈ { S 0 , S ′ } , resulting in a contradiction. Therefore, homoclinic orbits are non-existent in system (2.1).

(c) In light of (a), one-dimensional branch of W + u ( S 0 ) (resp. W − u ( S 0 ) ) has an ω -limit, i.e., q ∈ { S 0 , S ′ } . Since V 1,2 ( S 0 ) > V 1,2 ( S ′ ) , q must be S ′ , resulting in the formation of one and only one heteroclinic orbit to S 0 and S ′ , as shown in Figures 7 and 8. The proof is complete.□

$Figure 7 When ( a , b ) = ( 1 , 2 ) \left(a,b)=\left(1,2) , (a) c = − 6 c=-6 , ( x 0 4 , y 0 4 , z 0 6 ) = ( 1.314 , 1.618 , 2.236 ) × 1 0 − 7 \left({x}_{0}^{4},{y}_{0}^{4},{z}_{0}^{6})=\left(1.314,1.618,2.236)\times 1{0}^{-7} , (b) c = 6 c=6 , ( x 0 5 , y 0 5 , z 0 6 ) = ( − 1.314 , − 1.618 , 2.236 ) × 1 0 − 7 \left({x}_{0}^{5},{y}_{0}^{5},{z}_{0}^{6})=\left(-1.314,-1.618,2.236)\times 1{0}^{-7} , an orbit heteroclinic to S 0 {S}_{0} and S ′ {S}^{^{\prime} } . (a) c = − 6 c=-6 and (b) c = 6 c=6 .$

Figure 7

When ( a , b ) = ( 1 , 2 ) , (a) c = − 6 , ( x 0 4 , y 0 4 , z 0 6 ) = ( 1.314 , 1.618 , 2.236 ) × 1 0 − 7 , (b) c = 6 , ( x 0 5 , y 0 5 , z 0 6 ) = ( − 1.314 , − 1.618 , 2.236 ) × 1 0 − 7 , an orbit heteroclinic to S 0 and S ′ . (a) c = − 6 and (b) c = 6 .

$Figure 8 When ( a , b ) = ( 1 , 3 ) \left(a,b)=\left(1,3) , (a) c = 6 c=6 , ( x 0 4 , y 0 4 , z 0 6 ) = ( 1.314 , 1.618 , 2.236 ) × 1 0 − 7 \left({x}_{0}^{4},{y}_{0}^{4},{z}_{0}^{6})=\left(1.314,1.618,2.236)\times 1{0}^{-7} , (b) c = − 6 c=-6 , ( x 0 5 , y 0 5 , z 0 6 ) = ( − 1.314 , − 1.618 , 2.236 ) × 1 0 − 7 \left({x}_{0}^{5},{y}_{0}^{5},{z}_{0}^{6})=\left(-1.314,-1.618,2.236)\times 1{0}^{-7} , an orbit heteroclinic to S 0 {S}_{0} and S ′ {S}^{^{\prime} } . (a) c = 6 c=6 and (b) c = − 6 c=-6 .$

Figure 8

When ( a , b ) = ( 1 , 3 ) , (a) c = 6 , ( x 0 4 , y 0 4 , z 0 6 ) = ( 1.314 , 1.618 , 2.236 ) × 1 0 − 7 , (b) c = − 6 , ( x 0 5 , y 0 5 , z 0 6 ) = ( − 1.314 , − 1.618 , 2.236 ) × 1 0 − 7 , an orbit heteroclinic to S 0 and S ′ . (a) c = 6 and (b) c = − 6 .

4 Conclusion

In the sense of generalizing the second part of Hilbert’s renowned 16th problem, this study replaces the linear term c x of the Yang system with the ten-ninths-degree term c x 10 9 , and creates single-wing Lorenz-like attractors, asymmetric SDHC, single heteroclinic orbit, etc. Notably, the formation of single-wing Lorenz-like attractors may result from the collapse of these asymmetric SDHC, and that system casts a mirror image of these dynamics from − c to c .

Compared with the controlled Lorenz, Chen, and Lü systems with various strange attractors (i.e., asymmetric two-wing ones, partial ones, and single-wing ones), the newly reported one only generates single-wing ones. To some extent, this exotic phenomenon contributes to the broader understanding of Hilbert’s sixteenth problem, i.e., the degree also makes a real impact on the geometrical structure of strange attractors, except for the number and mutual disposition [6,7,11–13].

However, it is unclear whether there are hidden single-wing Lorenz-like attractors in that studied system or not. In what follows, we will continue to uncover the correction between the degrees and strange attractors, and provide reference for real world applications. The asymmetric single-wing attractors proposed here have promising applications in secure communication, where their asymmetric dynamics can enhance signal encryption robustness against symmetric attacks. In neural networks, the single-wing structure could model asymmetric information processing in biological systems, such as directional signal transmission in neural circuits. For system identification, stitching the right and left images of single-wing Lorenz-like attractor with ± c together complicates the problem of discovering governing equations [18].

Funding information: This work was supported in part by Zhejiang Public Welfare Technology Application Research Project of China under Grant LGN21F020003, and in part National Natural Science Foundation of China under Grant 12001489. Furthermore, the authors sincerely thank the anonymous editors and reviewers for their meticulous reading and constructive feedback, which have significantly enhanced the quality of this manuscript.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: All data generated or analyzed during this study are included in this published article.

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Received: 2025-03-03

Revised: 2025-04-13

Accepted: 2025-05-08

Published Online: 2025-06-05

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2025-0165

Keywords for this article

heteroclinic orbit; generalization of the second part of Hilbert’s 16th problem; Lyapunov function; single-wing attractor; asymmetric singularly degenerate heteroclinic cycle

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