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A proof for a theorem on intertwining property of attraction basin boundaries in planar dynamical systems
Chaos, Solitons and Fractals 15 (2003) 655–657 www.elsevier.com/locate/chaos
A proof for a theorem on intertwining property of attraction basin boundaries in planar dynamical systems Xiao-Song Yang Institute for Nonlinear Systems, Chongqing University of Posts and Telecomm. Chongqing 400065, China Accepted 15 May 2002
Abstract In this paper we present a new rigorous proof to a theorem on intertwining property of attraction basin boundaries in planar systems discussed in the literature [Chaos, Solitons & Fractals 10 (1999) 1453]. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction In the last few decades, much research has been carried out on the structure of basin boundaries in dynamical system with more than one attractor coexisting (see [1–3] and references therein). Usually these dynamical systems are described by maps. If there are more than one attractors in such a system, then it is often the case that a wild topology such as Wada lake phenomenon [4] takes place on boundaries of attraction basins. A basin boundary is usually nonsmooth and likely to be fractal which makes predictability impossible due to the practical small uncertainties in initial conditions. As for a dynamical system generated via autonomous ordinary differential equations (ODEs), the basin boundary may still be intricate in the case of the coexistence of several attractors. The basin boundary, although smooth, often winds or meanders, and the different basins of attraction with the winding boundary intertwine with each other. In addition, these basins become finer and finer and take a shape of snake-like thin bands. This phenomenon was observed in the work of Tedeschini-Lalli [5] without rigorous theoretical analysis. In [6] the authors tried to give a theorem concerning the so-called intertwining property of attraction basin boundaries. However the proof is not valid yet. The purpose of this note is to give a rigorous proof for the theorem discussed in [6]. For the convenience of discussions in next sections, some preliminaries are reviewed. Given a dynamical system with flow /t generated via differentiable ODEs x_ 1 ¼ V1 ðx1 ; x2 Þ;
x_ 2 ¼ V2 ðx1 ; x2 Þ
ð1Þ
defined onSR2 . Suppose A is an attractor of /t and N is an attracting neighborhood of A. Then the basin of attraction of A is B ¼ f/t ðN Þ : t 2 ð1; 0g, and the basin boundary is oB ¼ B B, where B denotes the closure of the basin B. Thus oB and B are closed invariant sets of /t . Furthermore, a sink is an attractive equilibrium point of /t which is stable in Lyopunov’s sense, a source is a repelling equilibrium point of /t , and a saddle point is an equilibrium point to which (locally) two orbits go as t ! 1 and other two orbits go as t ! 1 in its neighborhood (these notions can be found in any standard book on dynamical systems).
E-mail address: [email protected] (X.-S. Yang). 0960-0779/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 2 ) 0 0 1 5 4 - 6
656
X.-S. Yang / Chaos, Solitons and Fractals 15 (2003) 655–657
2. The theorem on intertwining of basin of attraction For two-dimensional ODEs with flow /t , the intertwining of basins of attraction can be defined as follows: Definition 2.1. Suppose that /t is a flow for an ODE on R2 . Two (or more than two) basins of attraction are said to be intertwined, if they have a common boundary and that common boundary, call it oB, has the following property: There are points x and y in oB such that for every e > 0, there exists t1 > 0 such that the point /ðy; t1 Þ is contained in the intersection of the e-disc Dðx; eÞ centered at x and a line Lx transversal to the vector field generated by the flow /t at the point x. Or equivalently Dðx; eÞ \ /ðy; t1 Þ \ Lx 6¼ Ø
ð2Þ
In this case system (1) is said to have the intertwining property. Remark 2.1. It is easy to see from this definition that if a two dimensional system has the intertwining property, then there exists a region in phase space where one fails to determine which basin of attraction a point with uncertainty belongs to, thus it is impossible to predict the dynamical behaviour of an orbit with initial condition in the intertwining region. The following is a restatement of the main theorem discussed in [6]. Theorem 2.1. Suppose system (1) is a structurally stable system generated via a vector field (ODEs) on R2 , which possesses three equilibrium points: two sinks A1 , A2 and a saddle point As whose unstable manifold connects the two sinks A1 and A2 . And the stable manifold of As is bounded as time t ! 1. Then (1) has the intertwining property. 3. Proof of the theorem The following fact is needed in our new proof: Lemma 3.1. Let q be a saddle point of (1), W s ðqÞ and W u ðqÞ are its stable and unstable manifold. If its unstable manifold approaches a asymptotically stable equilibrium point e, then W s ðqÞ is a part of the stability boundary of e. Proof. Denote by oB the basin boundary of e, we will prove that W s ðqÞ oB. Let W u ðqÞ W u ðqÞ is the branch that tends to the equilibrium point e. Consider a small segment of line C that belongs to the basin B of e and has a transversal intersection point with W u ðqÞ . In view of well known k-Lemma [7], there exists a small ball Dðq; dÞ with center q and radius d, such that for every p 2 Dðq; dÞ \ ðW s ðqÞ qÞ, and a ball Dðp; eÞ with center p and arbitrary small radius e, there is a sufficiently large T > 0 and for t > T , /ðt; CÞ \ Dðp; eÞ 6¼ Ø: It follows that Dðp; eÞ contains points of basin B of e due to invariance of B, therefore p is contained in the closure of basin B of e. However, p does not approach e, therefore p 2 oB, consequently Dðq; dÞ \ W s ðqÞ B. Because of invariance of the basin boundary, it is easy to see that W s ðqÞ oB. Proof of the theorem. Denote by W1s and W2s the two branches of the stable manifold of As (see Fig. 1). They are clearly a part of the boundary of the basins of A1 and A2 in view of Lemma 3.1. Consider a point p 2 W1s fAs g. Then the orbit /ðp; tÞ does not approach As as t ! 1, because (1) is structurally stable and consequently has no homoclinic orbit. In addition, it is apparent that /ðp; tÞ does not go to any sinks as t ! 1. Therefore its alpha-set aðpÞ contains no equilibrium point, which means that aðpÞ is a periodic orbit, i.e., a simple closed curve in view of the hypothesis that the stable manifold of As is bounded as time t ! 1. Now consider (1) on the bounded region C with boundary aðpÞ. From the above arguments, it can been seen that W1s approaches aðpÞ backwards as t ! 1. As for W2s , its alpha-set aðp0 Þ (p0 2 W2s fAs g) is likewise a periodic orbit which surrounds the equilibrium points in C, and the uniqueness of solution guarantees that aðpÞ ¼ aðp0 Þ. Consider a line L intersecting aðpÞ transversely as in Definition 2.1. It can be asserted that W1s and W2s backwards intersect L alternatively infinite many times as t ! 1. If this is not the case, without loss of generality, it can be supposed that W1s backwards intersect L successively at x1 and x2 , and W2s does not intersect L in the segment Lx1 x2 ended by x1 and x2 . Then consider the set
X.-S. Yang / Chaos, Solitons and Fractals 15 (2003) 655–657
657
Fig. 1. Two attractors with their attraction basins intertwining each other.
F ¼
[
f/ðx1 ; tÞ : t1 6 t 6 0g
[
Lx1 x2 ;
where /ðx1 ; 0Þ ¼ x1 , /ðx1 ; t1 Þ ¼ x2 and obviously t1 < 0. Clearly F is a closed curve that bounds a region containing the saddle point As and in terms of the uniqueness of solution, W2s cannot remain in this region and will go out of it only through Lx1 x2 as t ! 1, thus contradicting to the assumption that W2s does not intersect Lx1 x2 . From the arguments above it is easy to see that W1s and W2s satisfy (2) in Definition 2.1, and therefore (1) has the intertwining property.
Acknowledgement This work is partially supported by the Applied Science Foundation of Chongqing City Scientific Community.
References [1] [2] [3] [4]
Kennedy J, York JA. Basins of wada. Physica D 1991;51:213–25. McDonald SW, Grebogi C, Otte E, York JA. Fractal basin boundaries. Physica D 1985;17:125–53. Nusse HE, Otte E, York JA. Saddle-node bifurcation on fractal basin boundaries. Phys Rev Lett 1995;75:2483–5. El Naschie MS. Wild topology, hyperbolic geometry and fusion algebra of high energy particle physics. Chaos, Solitons & Fractals 2002;13:1935–45. [5] Tedeschini-Lalli L. Smoothly intertwined basins of attraction in a prey-predator model. Acta Applicandae Mathematicae 1995;38:139–41. [6] Yang X-S, Duan CK. Intertwined basin of attraction in systems of ODE. Chaos, Solitons & Fractals 1999;10:1453–6. [7] Palis J, De Melo Jr W. Geometry theory of dynamical systems: an introduction. New York: Springer-Verlag; 1981.
A proof for a theorem on intertwining property of attraction basin boundaries in planar dynamical systems Xiao-Song Yang Institute for Nonlinear Systems, Chongqing University of Posts and Telecomm. Chongqing 400065, China Accepted 15 May 2002
Abstract In this paper we present a new rigorous proof to a theorem on intertwining property of attraction basin boundaries in planar systems discussed in the literature [Chaos, Solitons & Fractals 10 (1999) 1453]. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction In the last few decades, much research has been carried out on the structure of basin boundaries in dynamical system with more than one attractor coexisting (see [1–3] and references therein). Usually these dynamical systems are described by maps. If there are more than one attractors in such a system, then it is often the case that a wild topology such as Wada lake phenomenon [4] takes place on boundaries of attraction basins. A basin boundary is usually nonsmooth and likely to be fractal which makes predictability impossible due to the practical small uncertainties in initial conditions. As for a dynamical system generated via autonomous ordinary differential equations (ODEs), the basin boundary may still be intricate in the case of the coexistence of several attractors. The basin boundary, although smooth, often winds or meanders, and the different basins of attraction with the winding boundary intertwine with each other. In addition, these basins become finer and finer and take a shape of snake-like thin bands. This phenomenon was observed in the work of Tedeschini-Lalli [5] without rigorous theoretical analysis. In [6] the authors tried to give a theorem concerning the so-called intertwining property of attraction basin boundaries. However the proof is not valid yet. The purpose of this note is to give a rigorous proof for the theorem discussed in [6]. For the convenience of discussions in next sections, some preliminaries are reviewed. Given a dynamical system with flow /t generated via differentiable ODEs x_ 1 ¼ V1 ðx1 ; x2 Þ;
x_ 2 ¼ V2 ðx1 ; x2 Þ
ð1Þ
defined onSR2 . Suppose A is an attractor of /t and N is an attracting neighborhood of A. Then the basin of attraction of A is B ¼ f/t ðN Þ : t 2 ð1; 0g, and the basin boundary is oB ¼ B B, where B denotes the closure of the basin B. Thus oB and B are closed invariant sets of /t . Furthermore, a sink is an attractive equilibrium point of /t which is stable in Lyopunov’s sense, a source is a repelling equilibrium point of /t , and a saddle point is an equilibrium point to which (locally) two orbits go as t ! 1 and other two orbits go as t ! 1 in its neighborhood (these notions can be found in any standard book on dynamical systems).
E-mail address: [email protected] (X.-S. Yang). 0960-0779/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 2 ) 0 0 1 5 4 - 6
656
X.-S. Yang / Chaos, Solitons and Fractals 15 (2003) 655–657
2. The theorem on intertwining of basin of attraction For two-dimensional ODEs with flow /t , the intertwining of basins of attraction can be defined as follows: Definition 2.1. Suppose that /t is a flow for an ODE on R2 . Two (or more than two) basins of attraction are said to be intertwined, if they have a common boundary and that common boundary, call it oB, has the following property: There are points x and y in oB such that for every e > 0, there exists t1 > 0 such that the point /ðy; t1 Þ is contained in the intersection of the e-disc Dðx; eÞ centered at x and a line Lx transversal to the vector field generated by the flow /t at the point x. Or equivalently Dðx; eÞ \ /ðy; t1 Þ \ Lx 6¼ Ø
ð2Þ
In this case system (1) is said to have the intertwining property. Remark 2.1. It is easy to see from this definition that if a two dimensional system has the intertwining property, then there exists a region in phase space where one fails to determine which basin of attraction a point with uncertainty belongs to, thus it is impossible to predict the dynamical behaviour of an orbit with initial condition in the intertwining region. The following is a restatement of the main theorem discussed in [6]. Theorem 2.1. Suppose system (1) is a structurally stable system generated via a vector field (ODEs) on R2 , which possesses three equilibrium points: two sinks A1 , A2 and a saddle point As whose unstable manifold connects the two sinks A1 and A2 . And the stable manifold of As is bounded as time t ! 1. Then (1) has the intertwining property. 3. Proof of the theorem The following fact is needed in our new proof: Lemma 3.1. Let q be a saddle point of (1), W s ðqÞ and W u ðqÞ are its stable and unstable manifold. If its unstable manifold approaches a asymptotically stable equilibrium point e, then W s ðqÞ is a part of the stability boundary of e. Proof. Denote by oB the basin boundary of e, we will prove that W s ðqÞ oB. Let W u ðqÞ W u ðqÞ is the branch that tends to the equilibrium point e. Consider a small segment of line C that belongs to the basin B of e and has a transversal intersection point with W u ðqÞ . In view of well known k-Lemma [7], there exists a small ball Dðq; dÞ with center q and radius d, such that for every p 2 Dðq; dÞ \ ðW s ðqÞ qÞ, and a ball Dðp; eÞ with center p and arbitrary small radius e, there is a sufficiently large T > 0 and for t > T , /ðt; CÞ \ Dðp; eÞ 6¼ Ø: It follows that Dðp; eÞ contains points of basin B of e due to invariance of B, therefore p is contained in the closure of basin B of e. However, p does not approach e, therefore p 2 oB, consequently Dðq; dÞ \ W s ðqÞ B. Because of invariance of the basin boundary, it is easy to see that W s ðqÞ oB. Proof of the theorem. Denote by W1s and W2s the two branches of the stable manifold of As (see Fig. 1). They are clearly a part of the boundary of the basins of A1 and A2 in view of Lemma 3.1. Consider a point p 2 W1s fAs g. Then the orbit /ðp; tÞ does not approach As as t ! 1, because (1) is structurally stable and consequently has no homoclinic orbit. In addition, it is apparent that /ðp; tÞ does not go to any sinks as t ! 1. Therefore its alpha-set aðpÞ contains no equilibrium point, which means that aðpÞ is a periodic orbit, i.e., a simple closed curve in view of the hypothesis that the stable manifold of As is bounded as time t ! 1. Now consider (1) on the bounded region C with boundary aðpÞ. From the above arguments, it can been seen that W1s approaches aðpÞ backwards as t ! 1. As for W2s , its alpha-set aðp0 Þ (p0 2 W2s fAs g) is likewise a periodic orbit which surrounds the equilibrium points in C, and the uniqueness of solution guarantees that aðpÞ ¼ aðp0 Þ. Consider a line L intersecting aðpÞ transversely as in Definition 2.1. It can be asserted that W1s and W2s backwards intersect L alternatively infinite many times as t ! 1. If this is not the case, without loss of generality, it can be supposed that W1s backwards intersect L successively at x1 and x2 , and W2s does not intersect L in the segment Lx1 x2 ended by x1 and x2 . Then consider the set
X.-S. Yang / Chaos, Solitons and Fractals 15 (2003) 655–657
657
Fig. 1. Two attractors with their attraction basins intertwining each other.
F ¼
[
f/ðx1 ; tÞ : t1 6 t 6 0g
[
Lx1 x2 ;
where /ðx1 ; 0Þ ¼ x1 , /ðx1 ; t1 Þ ¼ x2 and obviously t1 < 0. Clearly F is a closed curve that bounds a region containing the saddle point As and in terms of the uniqueness of solution, W2s cannot remain in this region and will go out of it only through Lx1 x2 as t ! 1, thus contradicting to the assumption that W2s does not intersect Lx1 x2 . From the arguments above it is easy to see that W1s and W2s satisfy (2) in Definition 2.1, and therefore (1) has the intertwining property.
Acknowledgement This work is partially supported by the Applied Science Foundation of Chongqing City Scientific Community.
References [1] [2] [3] [4]
Kennedy J, York JA. Basins of wada. Physica D 1991;51:213–25. McDonald SW, Grebogi C, Otte E, York JA. Fractal basin boundaries. Physica D 1985;17:125–53. Nusse HE, Otte E, York JA. Saddle-node bifurcation on fractal basin boundaries. Phys Rev Lett 1995;75:2483–5. El Naschie MS. Wild topology, hyperbolic geometry and fusion algebra of high energy particle physics. Chaos, Solitons & Fractals 2002;13:1935–45. [5] Tedeschini-Lalli L. Smoothly intertwined basins of attraction in a prey-predator model. Acta Applicandae Mathematicae 1995;38:139–41. [6] Yang X-S, Duan CK. Intertwined basin of attraction in systems of ODE. Chaos, Solitons & Fractals 1999;10:1453–6. [7] Palis J, De Melo Jr W. Geometry theory of dynamical systems: an introduction. New York: Springer-Verlag; 1981.