On a Method of Finding Homoclinic and Heteroclinic Orbits in. Multidimensional Dynamical Systems

Transcription

1 Applie Mathematics & Information Sciences 4(3) (), An International Journal c Diie W Publishing Corporation, U. S. A. On a Metho of Fining Homoclinic an Heteroclinic Orbits in Multiimensional Dynamical Systems Gheorghe Tigan Politehnica University of Timisoara, P-ta Victoriei, Nr., 36, Timisoara, Romania Aress: gheorghe.tigan@mat.upt.ro Receive November, 8; Revise November 3, 8 The presence of boune orbits such as homoclinic an heteroclinic orbits are important in a ynamical system. Chaotical behavior or the presence of perioic orbits of a system is often precee by estruction of a homoclinic or heteroclinic orbit. In this work we give insights on a metho of etecting homoclinic or heteroclinic orbits in a three-imensional ynamical system. The metho can also be applie for multiimensional systems. Keywors: Dynamical systems, heteroclinic orbits, chaos. Mathematics Subject Classification: 37C9, 65P3. Introuction an Necessary Theoretical Notions In this paper we present insights on a metho ealing with etecting homoclinic or heteroclinic orbits in a ynamical system generate by ifferential equations. The iea is to trace the separatrices (stable an unstable manifols) until they hit a given surface an then look for conitions such that the separatrices meet one to another on this surface. Fining homoclinic an heteroclinic orbits in a given ynamical system is not an easy task but their presence tell us much about the behavior of the system. For eample, estroying a homoclinic or heteroclinic orbit can bring a system in chaos in some cases while in others it leas only to the appearance of perioic orbits. Consier a surface S in R 3 given by S : F (, y, z) =,F C (R 3 ). A normal vector to S at a point (, y, z) S is the graient vector graf (, y, z) =(F,F y,f z). The surface S splits the space in two regions accoring to F > (the positive region) or F < (the negative region) an assume the normal vector graf (, y, z) lies in the positive region.

2 384 Gheorghe Tigan If a trajectory (for t increasing ) of a ifferential system u = f(u), (.) where u =(ẋ, ẏ, ż), f(u) =(f (, y, z),f (, y, z),f 3 (, y, z)), hits the surface S at a point A(,y,z ) coming from the positive region F >, then the vector fiel f at A (which is tangent to the trajectory) makes with the graient vector graf at A an angle larger than 9, see Fig... Since cos(graf, f) = gra F f gra F f, it follows that gra F f<at A. On the other han, t F (, y, z) F = = F ẋ + F y ẏ + F z ż F = = graf f<, at A. gra F S f Figure.: An orbit tangent to the vector fiel f at a point A, entering transversally the surface S : F (, y, z) =coming from the positive region F>, makes with the graient vector graf at A an angle larger than 9. As a conclusion, we have Proposition.. If the erivative (for t increasing) along a trajectory of the system (.), t F (, y, z) F = <, then the trajectory hits the surface S : F (, y, z) =coming from the positive region F (, y, z) >. If t F (, y, z) F = >, then it crosses the surface S : F (, y, z) =coming from the negative region F (, y, z) <. For t ecreasing, the scenario is inversely, i.e. if t F (, y, z) F = <, the trajectory hits the surface S coming from the negative region an if t F (, y, z) F = >, the trajectory hits the surface S coming from the positive region.

3 Fining Homoclinic an Heteroclinic Orbits 385 Remark.. If t F (, y, z) F =, the surface S is calle without contact for trajectories of the system or a transversally section to the flow of the system. If t F (, y, z) F = = then the trajectory is inclue in S or tangent to it. Tracing the Unstable Manifol W u + We will apply the metho for etecting homoclinic orbits only but similarly it can be applie for heteroclinic orbits. The system consiere in this work is a Shimizu-Moriokalike 3D moel given by ẋ = y, ẏ =(a + )( z) ay, ż = z. (.) Applications of Shimizu-Morioka moels are suggeste in []. In [3, 4] two types of Lorenz-like attractors of this system are shown. Relationships with the theory of normal forms are reporte in [6, 8]. Computer assiste results on the behavior of the system are obtaine in [5]. The equilibrium points of the system are O(,, ),O (,, ),O (,, ). As the system is invariant uner the transformation, y y, its orbits are symmetrically with respect to the Oz ais, so, with no loss of generality, we may restrict our investigations to >. Since a homoclinic or heteroclinic orbit is boune an for z < the erivative ż>, they will lie in z>, so we may restrict further to z>. The Jacobian matri associate to the system at O(,, ) has the eigenvalues,, a an the corresponing eigenvectors (,, ), (,, ), ( /(a +),, ), so O(,, ) is a sale. Assume a + >. Therefore the system has a one-imensional unstable manifol W u an a two-imensional stable manifol W s passing through O(,, ). We have also the one-imensional tangent space to W u at O(,, ), TW u = {(, y, z) : = y, z =} an the two-imensional tangent space to W s at O(,, ), TW s = {(, y, z) :y + (a +)=,z R}. The Jacobian matri at O, has the characteristic polynomial p 3 +(a +)p + ap +a +=.

4 386 Gheorghe Tigan From Routh-Hourwitz conitions, the characteristic polynomial has all roots with negative real part if an only if a>. Notice that the positive part of Oz ais, enote by +, belongs to W s. Denote W u + the branch of W u lying in y> an W u the branch of W u lying in y<. Consier the equation of W u + in the neighborhoo of O(,, ) given by y = a + a + a z = b + b + b b (.) As W u = W+ W u u is invariant uner the action of the flow of the system, after computing the coefficients, equations of W+ u for small enough, are given by y = +,z= 3 +. (.3) So for small enough, W+ u lies in the region { } R = (, y, z): <y<, 3 <z<,>. In [7] we prove that W+ u lies in the region /3 <z< not only for small but for all >,y >. We want to estimate now the coorinate of the point where W+ u crosses y =. Consier the surface in R 3,y= b, <z<,b >. As b <,b> the surface y = b is below W+ u for small enough. We are intereste uner what conitions, trajectories lying in /3 <z< can cross this surface. Proposition.. The surface in R 3,y= b, <z<,b > is a surface without contact for trajectories of the system. If t increases, trajectories of the system cross the surface on the sie y< b b(a +3) for all << a ++b,b>. In particular, if b =we obtain that the trajectories of the system (an implicitly W+) u for <z< cross y =for. Proof. if b (a +3) < a ++b := u, we have ( y + b ) y= b > ( ( a ++b ) b (a +3) ) >. t We use z<. So the surface y = b, <z< is without contact for trajectories of the system an for t increases, trajectories of the system cut the surface on the sie

5 Fining Homoclinic an Heteroclinic Orbits 387 y< b for all << u,b >. Notice that u /b for all b an all a +>. Consequently, u =is the inferior (an maimum in this estimation) limit which is vali for all a +> such that trajectories cross the plane y =for >. Consier now a curve above W u +,y= c,c>,>. One may remark that for any >, c> we have c >for small enough, more eactly if <(c )/. So the surface y = c is above W u + for small enough. Proposition.. There eists c>,>, c/ = 3 such that the surface in R 3, y = c, 3 <z is a surface without contact for trajectories of the system for all >. Ift increases, trajectories of the system cross the surface for all a>,a 8a 8 > on the sie y>c. Consequently, for a>,a 8a 8 > trajectories (an implicitly W+) u starting in cross y =for 3. y<c, 3 <z Proof. We have ( y c + ) y=c (.4) t ( < ( + a + ) ) (a +3c) + c + ac a < 3 for all >if ( (a +3c) 4 + a + ) (c + ac a ), 3 or equivalently < 4(a ( +)(c ) (a + c +) ) := m. 3 (a c) +8(a +) As the intersection of y = c with O is an c/, we are intereste in the maimum of in orer to see how low on O trajectories can cross y =. So take = m an estimate the minimum of c/ for c>,a+>. Denoting p = c /, we obtain p c = 3 ( c +3ac 8a a 8 )( c ac +a + ) c 4 (a + c +) (a +)(c ). Further, p/ c =implies ac a c =with c = a + a 8a 8,c = a a 8a 8, a 8a 8

6 388 Gheorghe Tigan an 8a 3ac + a c +8=with c 3 = 3 4 a + 64a +7a +64,c 4 = a 64a +7a for any a +>. Case. If a 8a 8 > an a>then we have c 4 < <c <c 3 <c. Evaluating the sign of p/ c for c on (, + ) we get that c an c are the minimum points. Evaluating also p at c an c we get surprisingly, p(c )=p(c )=3, for any a>,a 8a 8 >. So u := c/ = 3 is the limit for a>,a 8a 8 >, with the property that trajectories can cross y= only for < u. Case. If a 8a 8 < then c, are not real anymore an c 4 < < c 3 for any a +>. So the minimum in this case is c = c 3 an one can show that the minimum of u = c/ is also 3. Case 3. If a + >,a < an a 8a +8 > then c, are real again but c, <. We keep still c 4 < <c 3 for any a +> an again c = c 3 is the minimum. With these constraints of a, u = c/ is ecreasing with respect to a, raging from + to 3. So these two last cases o not offer a better option for u. Remark.. The above Propositions.-. say that, the separatri W u + crosses the plane y =in a point K( u,,z u ) with < u < 3 if a is large enough (in fact for a 8a 8 > an a>). 3 Tracing the Stable W s Manifol As the positive part of the Oz ais, enote +, belongs to W s, we epress W s in a small neighborhoo of some parts of + in the form y = f(t), z = z e t,t >. Denote TW s A the tangent space to W s at the point A. In [7] we prove the following assertions. Proposition 3.. a) There eists a sequence z,z,...,z k,..., z k+ >z k,z > + a 4(a +)

7 such that the tangent space Fining Homoclinic an Heteroclinic Orbits 389 TW s Ak = {(, y, z) :y =}, where k =,,,...an A k =(,,z k ). b) For z<z we epress TW s A = {(, y, z) :y = f (t), z = z e t,t > }, where A =(,,z) an f (t) is a solution of the Riccati equation (generally unsolvable) f (t) =f (t) af(t) (a + )( z e t ), (3.) which is boune for t>,f () = an such that (a + ) (a +) 4(a +)z e t <f (t) <a+,t>t >, (3.) where t := ln 4(a +)z (a +) >. This result implies that in the neighborhoo of the integral line + : { = y =,z > }, the two-imensional manifol W s for z z can be epresse as So for <ε we can fin a curve {y = f (t), z = z e t,t }. C = W s { + y = ε, z z }. Proposition 3.. Consier the region in R 3,R = { (a +)<y<} where y = an y = (a+) are seen as surfaces in R 3. Then the curve C R for ε small enough. In aition, for ecreasing t, each trajectory starting on C up to its intersection with the plane y =, remains in the region R an will cross this plane only for z>. Proof. It is clear that the curve C lies in R because >,f (t) >,t > so y< an from f (t) <a+,t > we get y +(a +)>. Recall that TW s = {(, y, z) :y + (a +)=,z R}. We have also t (y +(a +)) y= (a+) = (a +)z for all z. Hence, the surface y +(a +) =is a surface without contact for trajectories of the system an the trajectories, for t ecreasing, cross the surface on the sie y +(a +)< for any z >. From t y y= =(a + )( z) < if >,z >, we have that, for t ecreasing, trajectories can cross y =on the sie y< only if >,z >. Consier in the following a surface in R 3 given by y = b c, b >,c>a+with z>. As b c < (a +) for small enough, this surface is below W s for small enough. Evaluate, t (y b + c) y=b c = ( z a ac + az + ab 3bc + c +b ) < (3.3)

8 39 Gheorghe Tigan as long as > an b + b(a 3c) + z a ac + az + c >. (3.4) Relation (3.4) hols true if an only if b ( 8a 8z +ac 8az + a + c +8 ) <, that is, if z > (8a +ac + a + c +8)/(8 + 8a). As the minimum of (8a +ac + a + c +8)/(8+8a) for c > a + > is attaine at c = a, if < a < / we can choose c = a > a + >. This leas to z >. So if < a < /, there eists c = a > a +such that t (y b + c) y=b c < for any >,z > an for any b >. Therefore, the surface in R 3,y= b + a, b >, <a< / with z>, z,> is a surface without contact for trajectories of the system an the trajectories for t ecreasing cross this surface for z>on the sie y<b + a. As the curve C contains points below z =, some trajectories starting on it may cross the surface y = b + a before meeting z =, so for such trajectories we have no bounary. But, take b = a >. The nonzero intersection of the curve y = a + a with the O ais is =. Make now the following assumption: Assumption A. Assume that a trajectory lying in z >, starting on C an remaining in z< before to cross the surface y = a + a (if enters z>, it can not cross the surface anymore), will cross first the surface z =. With this assumption at han, such a trajectory after crossing the surface z =,gives z(t) ecreasing an (t) increasing (ż>, ẋ<), that is, the trajectory remains in z< an it can not cross y =(see Prop. 3.) so it may escape on the surface z =. As the trajectories starting in y> a + a, > (in particular those starting on C) can not cross y = a + a for any >,z > they will cross y =in a point (,,z) such that <,z >. A trajectory, for t ecreasing, starting on C at a point such that z > increases in y(t) because ẏ <. Also it increases in (t) an in z(t) as ẋ<, ż<so it will cross surely the plane y =. It woul ecrease in z(t) after crossing the plane z = which is not possible because the surface z =, (, ) is below the surface z =, (, ). If / a, we can not fin anymore c an b such that t (y b + c) y=b c <, so we only can infer that the curve C lies in z>in this case. Summarizing, we have the following useful intuitive conjecture. We can not give it as a result since the trajectory may hit the surface y = a + a first an then z =. Conjecture. Consier <a. Then for t ecreasing an <a< /, the

9 Fining Homoclinic an Heteroclinic Orbits 39 trace of W s through the flow of the system on the plane y =is a curve lying completely in <, z> an for / a lying in z>. The evolution of the curve C through the flow of the system for a is summarize in the following result. Theorem 3.. If a> the trace of the curve C through the flow of the system on the plane y =, is a curve Γ joining A (,,z ) with a point B (,, ), lying on the curve z =, an a point B (,, ), > z lying on z =. Proof. As a>we have z >. For t ecreases, trajectories starting on C for z> an for S := (a+)( z) ay < will increase in all three irections as ẋ<, ẏ<, ż< an some of them will cross the plane y =an others coul meet the surface z =. Consiering z arbitrarily fie, the surface S is generate by the lines y = m with the slope m =(a + )( z)/a, see Fig. 3.. On the other han, if S>then they will ecrease in y(t), an if not meet S =they will cross the surface z =. So we obtaine a curve Γ joining the points A,B an O which belongs also to W s. The part of the curve Γ which oes not lie in y =an which is above the plane z =is further translate through the flow of the system in a curve Γ on the plane y =. Recall that from Proposition 3. trajectories can cross the plane y =only for z>. The curve Γ ens in y =,z = because at B (,, ) the trajectory is tangent to the plane y =as ẏ y=,z= =an after the tangency it returns to the region where y<if >or crosses the plane y = if <. If =it remains there as (,, ) is an equilibrium point. A similar argument is use in [] where also is prove that the point B is unique. 4 z y 4 Figure 3.: Crossing the surface S given by y =(a+)( z)/a, the flow of the system approaches or eparts the plane y =. Let us now escribe the scenario leaing to a homoclinic orbit consiering the above Conjecture is vali. Denotes in the following by D + the region on the plane y = efine by the curve Γ an the lines z =,z= z,=an by D the remaining region in the strip <z<z,>.

10 39 Gheorghe Tigan Define the function g(a) = min,z ( u, Γ ) i(c), where u is the O-coorinate of the point P ( u,,z u ) where W+ u crosses the plane y =, i(c) is + if the point P lies in D + an otherwise an (, y) is the Eucliian istance in the plane. From Conjecture an Theorem 3. we have that the function g(a) is continuous with respect to the parameter a for a +> an from Conjecture an from Proposition. we have in aition g(a) < for a (, /). For a>, a 8a 8 > an again from Theorem 3. we get that the intersection of the curve Γ with the region on y =, given by >z,z>is a curve entirely containe in > 3 because > z > + a 4(a+) while from Proposition. we have that P satisfies u < 3 so P D + an g(a) >. Therefore, there eists a number a ( /, 8.9) such that g(a )=. Summarizing, we have the range of the parameter a where the system can have a homoclinic loop. Conjecture. There eists a value a ( /, 8.9) of intersection of the curve Γ with W+ u such that, the system (.) corresponing to this value, has a homoclinic loop to the sale point O(,, ). Numerical investigations reveal inee a unique point a.78 such that the system possesses a homoclinic orbit. In Fig. 3. is presente a single homoclinic orbit lying in >,y >,z > while Fig. 3.3 isplays two symmetrical homoclinic orbits to O(,, ) corresponing to a. y.5 z.5 Figure 3.: One homoclinic orbit of the system for a. 3 y.5 z.5.5 Figure 3.3: Two homoclinic orbits of the system for a.

11 Fining Homoclinic an Heteroclinic Orbits z.5 z.5.5 Figure 3.4: Separatrices of the system for a =(left) an a =(right). z 6 z Figure 3.5: Orbits of the system for a =. (left) an a =.5 (right). 4 Conclusions Fining out close orbits in a multiimensional ynamical system is a challenging problem. Of this type of orbits, homoclinic an heteroclinic ones play an important role in unerstaning the behavior of the system. Uner some conitions, homoclinic or heteroclinic bifurcations lea to perioic orbits or chaos in a system. We pointe out here etails on a metho of etecting homoclinic orbits in a three-imensional ynamical system. The metho is applicable for higher-orer ynamical systems. Tracing the separatrices an their intersections with a given surface, we showe that there are conitions for them to meet one to another. Their intersections imply the eistence of a homoclinic orbit. While in this paper we registere some analytical results, others we ha to left as conjectures. Nevertheless, numerical results offer a goo basis for these conjectures. In [7] we starte to improve these results. References [] V. N. Belykh, Bifurcations of separatrices of a sale point of the Lorenz system, Differential Equations (984), [] A. Rucklige, Poster Nato Av. Res. Workshop in Brussels (99). [3] A. L. Shilnikov, Bifurcation an chaos in the Marioka-Shimizu system, Sel. Math. Sov. (99), 5 7. [4] A. L. Shilnikov, Methos of Qualitative Theory an Theory of Bifurcatios, Gorky St. Univ. (989), 3 38.

12 394 Gheorghe Tigan [5] A. L. Shilnikov, On bifurcations of the Lorenz attractor in the Shimizu-Morioka moel, Physica D 6 (993), [6] A. L. Shilnikov, L. P. Shilnikov, an D. V. Turaev, Normal forms an Lorenz attractors, IJBC 3 (993), [7] D. Turaev an G. Tigan, Analytical search for homoclinic bifurcations in a Shimizu- Morioka moel, in preparation. [8] A. Vlaimirov, G. Volkov, D. Yu, Technical Digest Series, Opt. Soc. of America (99), 98. Gheorghe Tigan is a Lecturer in the Department of Mathematics, Politehnica University of Timisoara, Romania. He obtaine his Ph.D. in 6 from the Department of Mathematics, West University of Timisoara, Romania. His scientific interest is in the Dynamical Systems Theory.