- Email: [email protected]
What is the simplest dissipative chaotic jerk equation which is parity invariant?
3 January 2000
Physics Letters A 264 Ž2000. 383–389 www.elsevier.nlrlocaterphysleta
What is the simplest dissipative chaotic jerk equation which is parity invariant? J.-M. Malasoma
)
ENTPE-DGCB, URA CNRS 1652, Vaulx en Velin 69518, France Received 14 September 1999; received in revised form 10 November 1999; accepted 11 November 1999 Communicated by C.R. Doering
Abstract We investigate the chaotic dynamics of an autonomous scalar third-order differential equation. This system seems to be the algebraically simplest and previously unknown example of a dissipative chaotic jerky flow which is parity invariant. It displays chaotic behaviours in two distinct ranges of its control parameter. Deterministic chaos is principally observed from a symmetric limit cycle which after a symmetry-breaking bifurcation gives rise to two cascades of flip bifurcation. Then two coexisting asymmetric chaotic attractors are observed, and after a symmetry-restoring crisis a symmetric chaotic attractor is created. Chaotic attractors also coexist in another very narrow range of control parameter as results of two period doubling cascades of bifurcation from a pair of mutually symmetric coexisting limit cycles. q 2000 Elsevier Science B.V. All rights reserved. PACS: 05.45.Ac; 05.45.-a; 02.30.Hq; 02.60.Cb Keywords: Jerk; Chaos; Attractor; Differential equation; Parity
Recently, some papers w1–5x appeared in response to an exciting question posed by Gottlieb w6x concerning existence of simple jerk equations which may exhibit a chaotic behaviour. A jerk equation is an autonomous third-order differential equation of the form ...
x s j Ž x , x˙ , x¨ .
Ž 1.
Here, x denotes a scalar, for example the position coordinate, the overdot represents the time derivative
)
E-mail: [email protected]
and the jerk function j is the time derivative of the acceleration w7x. The Poincare–Bendixson theorem shows that ´ chaos does not exist in a two-dimensional autonomous system. Since Eq. Ž1. can equivalently be written as three, first-order, ordinary differential equations, a nonlinear jerk function j is a necessary conditions for deterministic chaos. Moreover it is well known that polynomials are the algebraically simplest smooth functions. Therefore, given a general propriety Ž P . for example the type of the nonlinearity: quadratic or cubic etc., we consider the set of the jerk equations which verify Ž P . and have jerk functions which are polynomials with only one alge-
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 8 1 9 - 1
384
J.-M. Malasomar Physics Letters A 264 (2000) 383–389
braic nonlinearity and the minimal number of terms that allow chaotic behaviour. We shall call this set the class of the simplest chaotic jerk equation with the propriety Ž P .. For example let us consider dissipative jerk equations with only quadratic nonlinearities. After an exhaustive numerical investigation of such equations, Sprott found what seems to be the class of the simplest dissipative quadratic jerk equations which exhibit chaotic behaviour w1,2x. This class contains two members, but these equations are equivalent to within a constant. Each equation has a multivariate polynomial jerk function with only three terms including a single quadratic nonlinearity. Symmetry has always played an important role in physics, from fundamental formulations of basic principles to concrete applications, and is present in a variety of chaotic systems. This fact raises a natural question: What is the class of algebraically simplest dissipative jerk equations which are invariant under the parity transformation x yx and that exhibit chaotic behaviour? The problem is to find multivariate polynomial functions j with a single algebraic odd nonlinearity and the minimum number of terms that give dissipative chaotic systems of the form Ž1.. Since cubic nonlinearities are the simplest among odd algebraic nonlinearities we will focus on this case. Examples of such dissipative polynomial jerk equations with only one cubic nonlinearity and chaotic dynamics, are already known. The older of them is the almost forgotten thermal convection model of Moore and Spiegel proposed only three years later than the popular Lorenz model w8x. This system can be written in the nondimensional form
™
...
x q x¨ q Ž T y R q Rx 2 . x˙ q Tx s 0
where R and T are parameters. Solutions to this equation are thought to be chaotic when R s 100 and T s 20 but also in an entire region of the R–T plane w8–10x. Later, Auvergne and Baglin w11x described, in a rough approximation, the motion of an ionization zone in a star by an equation of the same type: ...
x q x¨ q l Ž 1 q m x 2 . x˙ q n x s 0
where the coefficients are given by the following expressions
ls
g1 A2 K 2
,
ms
g2 Ž1 q g1 . g1
2
,
ns
B
1
A A2 K 2
The term A2 K 2 gives a measure of the nonadiabaticity and the term ArB can be considered as a source term of the star. Chaotic behaviour occurs for example with g 1 s 0.3, g 2 s 25.0, A2 K 2 s 0.01 and BrA s 0.12 or BrA s 0.022. It is also the case of the Arneodo–Coullet–Spie´ gel–Tresser polynomial jerk equation ...
x q Ax¨ q Bx˙ q Cx q Dx 3 s 0
which exhibits chaotic dynamics for various values of the four parameters w12–14x. We can use for example the following sets A s 1, B s 3.5,C s 9.6, D s y14 or A s 1, B s 3.5,C s y5.5, D s 14 . These examples demonstrate the existence of polynomial jerk functions with four terms including a single cubic nonlinearity, for which Eq. Ž1. has chaotic solutions. In a more recent paper w1x, Sprott reported the results of an almost systematic numerical examination of third-order, one-dimensional, autonomous, ordinary differential equations with cubic nonlinearities and chaotic dynamics. Four functionally distinct forms, not given in his paper, were found with four terms and one cubic nonlinearity. But surprisingly no cases as simple as the quadratic nonlinearity, i.e. with three terms including a single nonlinearity, were found by this method. The aim of this Letter is to give such an example and also to analyze with some details its chaotic dynamics. The system to be considered is the following ...
x q a x¨ y xx˙ 2 q x s 0
Ž 2.
where the parameter a is restricted to be positive in order to have a dissipative flow. Eq. Ž2. can be derived from the Newtonian equation: x¨ s ya x˙ q
t
Ht x Ž t .
x˙ 2 Ž t . y 1 dt
0
governing the one-dimensional motion of point particle of mass unit under the influence of a force that depends on the instantaneous velocity Žviscous
J.-M. Malasomar Physics Letters A 264 (2000) 383–389
damping. but also on the position and velocity history. Eq. Ž2. has a single fixed point at the origin of the phase space which eigenvalues satisfy the characteristic equation P Ž l. s l3 q al2 q 1 s 0. An elementary study proves that this polynomial has only one real root lr - y2 ar3 which is therefore negative. Since the characteristic equation is a cubic equation with real coefficients, we will have without loss of generality P Ž l. s Ž l y l r .Ž l y l c .Ž l y lc . where lc is a complex number. After expanding the above equation and comparing the coefficients with those of the original characteristic equation we come up with the relation ReŽ l c . s 2y1ly2 r . We conclude that the real part of lc is positive and that the fixed point is unstable. The negative real eigenvalue is associated with a one-dimensional stable manifold, whereas both complex conjugate eigenvalues, with positive real parts, are associated with a two-dimensional unstable manifold in which trajectories are spiraling outwards. Thus the origin of the phase space is a saddle-focus with an instability index of 2. We note that Eq. Ž2. is invariant under the parity transformation
Ž x , y, z .
385
For a ) 2.540 all solutions numerically found are unbounded. At the contrary for a Q 2.540, depending on initial conditions, solutions of Eq. Ž2. are either unbounded or they settle down on a symmetric stable limit cycle. This stable solution has been studied via a Poincare´ section S defined by
S s Ž x , z . g R 2 N y s 0, y˙ ) 0 4
Ž 3.
Its basin of attraction appears to be very small and Fig. 1 shows its trace with the plane S for a s 2.5. We can notice that this section is obviously parity invariant. The basin exhibits also three spiraling structures around the fixed point at origin of the phase space and around the two symmetric points Ž"3.81,0,. 25.08.. For the same value a s 2.5 simple initial conditions which belong to this basin are given for example by Ž"4,0,0.. When the control parameter reaches the value a s 2.196 . . . , this limit cycle loses stability in a symmetry-breaking bifurcation. Therefore, it becomes unstable and two coexisting asymmetric stable limit cycles are created. One is symmetric of the other. When the parameter a is further decreased
™ Ž yx ,y y,y z .
of the phase space R 3 s Ž x, y s x, ˙ z s x¨ .4. Therefore a solution of Eq. Ž2. that is invariant under this transformation is called a symmetric solution, whereas a solution that is not invariant under the same transformation is called an asymmetric solution. Consequently, for a specified value of the control parameter a , if an asymmetric solution is found, an other different asymmetric solution can be obtained by applying the above transformation. Hence, all asymmetric solutions occur in pairs. Scanning the range 0 - a - 5, in increments of 10y2 , we have analyzed numerical solutions for 10 6 randomly chosen initial conditions with N x N- 5, N x˙ N- 5, and N x¨ N- 5 for each value of a . The calculations were performed using a fourth-order Runge–Kutta algorithm with step size of D t s 10y3 . Since the most common dynamics involve unbounded solutions, we stop the numerical calculations whenever we detect the condition N x N qN x˙ N qN x¨ N) 10 5. When bounded solutions were found, the results were verified by using a 7–8 Runge– Kutta–Fehlberg adaptive step algorithm.
Fig. 1. Intersection with the plane x˙ s 0 of the basin of attraction of the symmetric stable limit cycle for the control parameter value a s 2.5. This section is clearly parity invariant. Three spiraling structures are clearly visible.
386
J.-M. Malasomar Physics Letters A 264 (2000) 383–389
both limit cycles lose together their stability through period doubling bifurcation at a s 2.105 . . . It is the beginning of two complete cascades of bifurcation that may be observed depending on the initial conditions and that lead to chaotic dynamics as illustrated by the bifurcation diagram shown in Fig. 2. In this figure, successive local minima of x, computed using Ž3. by the classical Henon trick w15x, ´ are recorded after transients have decayed, and plotted versus a . As a is decreased both attractors enlarge within their own basin of attraction. When the control parameter reaches the value a s 2.0644 . . . , they both simultaneously experience a tangency with the common basin boundary separating their two basins and a symmetry restoring crisis occurs. After the crisis, the asymptotic motion settles Fig. 3. Variations of the largest, non zero, Lyapunov exponent l m versus the parameter a in the same range as in Fig. 2. The onset of chaos is visible near a , 2.084, where l m first becomes positive.
down to a larger symmetric chaotic attractor. With further decrease of a this attractor enlarge within its basin of attraction and collides with the basin boundary at a s 2.0278 . . . It is the final boundary crisis which kills the symmetric chaotic attractor. The chaotic region falls in the range 2.0278 . . . - a 2.0840 . . . which is extraordinary tiny, and this is probably why this flow was missed during the Sprott’s earlier numerical search w1x. It is worth noticing that these two bifurcation diagrams are very similar to those obtain for a cubic map of the interval. For example let us consider the map fmŽ x . s ym x y x 3. The fixed point x s 0 is stable in the interval x y 1,1w and when m - y1 a pair of mutually symmetric equilibrium points x "s " y 1 y m which are stable in the domain x y 2,y 1w exist. At m s y1 two cascades of period doubling bifurcation develop when the parameter m decreases, cumulating in two asymmetric chaotic attractors. Then a symmetry-restoring crisis occurs at m s y2.599 and a symmetric chaotic attractor is created which is destroyed in a boundary crisis at m s y3. Traditionally, the Lyapunov exponents are used to distinguish a periodic solution from a chaotic behaviour. They have numerically been determined by
'
Fig. 2. Bifurcation diagram of successive local minima x min of x in the Poincare´ section Ž3. plotted versus control parameter a . The end of the cascade of period doubling bifurcation leading to chaos is clearly visible for both the coexisting asymmetric periodic solutions Žtop and bottom. when a decreases.
J.-M. Malasomar Physics Letters A 264 (2000) 383–389
a standard numerical method w16x for a in the same range as in Fig. 2. Over the intervals in which the system has stable periodic solutions, the Lyapunov exponents spectra takes the form Ž l1 , l2 , l3 . s Ž0,y,y ., while over the intervals in which the system exhibits chaos, the Lyapunov exponents spectra takes the form Ž l1 , l2 , l3 . s Žq,0,y .. Fig. 3 shows the variations of the largest, non zero, exponent l m versus parameter a . The onset of chaos is visible near a s 2.0840 . . . , where l m first becomes positive. For a - 2.0840 . . . the exponent l m generally increases, except for the dips caused by the windows of stable periodic behaviour. The most visible are the large dips due to the period-3 window near a , 2.043 and due to the period-5 windows near a , 2.0545 for the symmetric attractor. Also we can notice the period-3 window near a , 2.0743 for both asymmetric attractors. Fig. 4 shows a phase space portrait of the symmetric chaotic attractor for a s 2.05. The Lyapunov exponents are found to be in base-e: l1 s 0.0541, l2 s 0, and l3 s y2.1041. Therefore the corresponding Kaplan–Yorke dimension w17x is D KY s 2 y l1rl3 , 2.0257. Using the Poincare´ section S , a first return map is hereafter computed and displayed in Fig. 5 for a s 2.05. To a very good approximation all points appear to lie on a single curve and a cubic map constituted by three monotonic branches separated by two critical points given by C1 s y3.469 . . . and C2 s y3.088 . . . is obtained. The time series of x and its first three time-de... rivatives y s x, ˙ z s x¨ and j s x for a typical trajectory onto the attractor at a s 2.05 are displayed in Fig. 6. The average duration between two successive intersections of the trajectory with the Poincare´ sec-
Fig. 4. Phase space portrait of the symmetric chaotic attractor for the parameter value a s 2.05.
387
Fig. 5. First return map to the Poincare´ section S at a s 2.05. The map appears to be one-dimensional constituted by three monotonic branches separated by two critical points given by C1 sy3.469 . . . and C2 sy3.088 . . . .
tion S is found numerically to be T , 15.2. Therefore the contraction of the phase space per cycle is given by eTl 3 , 10y1 2 which helps to explain why the first return map of Fig. 5 looks like a one-dimensional map. We have also found another interval of the control parameter in which chaotic behaviour is present. For 0.0018 Q a , depending on initial conditions, the solutions of Eq. Ž2. are either unbounded or they settle down on two mutually symmetric coexisting stable
Fig. 6. Time series of x and its first three time-derivatives y s x, ˙ ... z s x¨ and js x for a typical trajectory onto the attractor displayed in Fig. 4 for a s 2.05.
388
J.-M. Malasomar Physics Letters A 264 (2000) 383–389
limit cycles. These solutions have been studied again via the Poincare´ section Ž3.. Their basins of attraction are extraordinary tiny with a very complex structure which is still under investigation. For example, simple initial conditions within these basins are Ž"5.0," 0.926," 0.15. when a s 0.07. When a is increased these cycles remain stable until a s 0.0724 . . . . Then they simultaneously lose their stability in two cascades of period doubling bifurcation leading to coexisting chaotic attractors. Beyond these cascades to chaos, the attractors develop as a is raised. After many band-merging bifurcations each diagram is constituted by four remaining bands. For one of these diagrams, these bands have separately been plotted in Fig. 7 in order to enlarge the vertical scale. Two interior crises w18x are visible which causes the sudden expansion of the chaotic bands at a , 0.075355 and at a , 0.07536. Fig. 8 shows the variations of the largest, non zero, exponent l m versus parameter a . The onset of chaos is visible near a s 0.0753514 . . . , where l m first becomes positive. The exponent l m globally increases until the final boundary crisis. Once again,
Fig. 8. Variations of the largest, non zero, Lyapunov exponent l m versus the parameter a in the same range as in Fig. 7. The onset of chaos is visible near a , 0.0753514, where l m first becomes positive.
it is worth noticing that these coexisting chaotic dynamics fall in the extraordinary small range 0.0753514 . . . - a - 0.0753624 . . . which is five
Fig. 7. Bifurcation diagram of successive local minima x min of x in the Poincare´ section Ž3. plotted versus control parameter a . Since after many band-merging bifurcations, the diagram is constituted by four bands, they have separately been plotted in order two enlarge the vertical scale.
J.-M. Malasomar Physics Letters A 264 (2000) 383–389
thousand times smaller than the case previously reported. In conclusion, we have shown that deterministic chaos exists in a polynomial dissipative jerk equation whose jerk function is constituted by only three terms including a single cubic nonlinearity. Moreover, the flow associated with this equation is parity invariant. Two different parameter ranges where a such behaviour exists have been found. The route to chaos is the standard cascade of period doubling. In both intervals we have observed locally coexisting chaotic attractors. Any polynomial jerk function with only three terms depends on three coefficients. By rescaling x and t it is always possible to normalize two of them. Therefore only one control parameter remains free. Our equation has the minimum number of terms that allows an adjustable control parameter and has only one cubic nonlinearity. Therefore it is unlikely that any algebraically simpler dissipative cubic polynomial form of jerky chaotic flows exists. We conjecture that Eq. Ž2. belongs to the class of the simplest dissipative chaotic jerk equations which are parity invariant. However it is difficult to rule out the possibility of finding new equations which belong to the same class.
389
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x
w18x
J.C. Sprott, Am. J. Phys. 65 Ž1997. 537. J.C. Sprott, Phys. Lett. A 228 Ž1997. 271. S.J. Linz, Am. J. Phys. 65 Ž1997. 523. S.J. Linz, Am. J. Phys. 66 Ž1998. 1109. R. Eichhorn, S.J. Linz, P. Hanggi, Phys. Rev. E 58 Ž1998. ¨ 7151. H.P.W. Gottlieb, Am. J. Phys. 64 Ž1996. R525. S.H. Schot, Am. J. Phys. 46 Ž1978. 1090. D.W. Moore, E.A. Spiegel, Astrophys. J. 143 Ž1966. 871. N.H. Baker, D.W. Moore, E.A. Spiegel, Q.J. Mech. Appl. Math. 24 Ž1971. 391. N.J. Balmforth, R.V. Craster, Chaos 7 Ž1997. 738. M. Auvergne, A. Baglin, Astron. Astrophys. 142 Ž1985. 388. P. Coullet, C. Tresser, A. Arneodo, Phys. Lett. A 72 Ž1979. ´ 268. A. Arneodo, P. Coullet, C. Tresser, Commun. Math. Phys. 79 ´ Ž1981. 573. A. Arneodo, P. Coullet, E.A. Spiegel, Phys. Lett. A 92 ´ Ž1982. 369. H. Henon, Physica D 5 Ž1982. 412. A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Physica D 16 Ž1985. 285. J.L. Kaplan, J.A. Yorke, in: H.-O. Peitgen, H.-O. Walther ŽEds.., Lecture Note in Mathematics, Vol. 730. Functional DifferentiaEquations and Approximations of Fixed Points, Springer, Berlin, 1979, p. 204. C. Grebogi, E. Ott, J.A. Yorke, Phys. Rev. Lett. 48 Ž1982. 1507.
Physics Letters A 264 Ž2000. 383–389 www.elsevier.nlrlocaterphysleta
What is the simplest dissipative chaotic jerk equation which is parity invariant? J.-M. Malasoma
)
ENTPE-DGCB, URA CNRS 1652, Vaulx en Velin 69518, France Received 14 September 1999; received in revised form 10 November 1999; accepted 11 November 1999 Communicated by C.R. Doering
Abstract We investigate the chaotic dynamics of an autonomous scalar third-order differential equation. This system seems to be the algebraically simplest and previously unknown example of a dissipative chaotic jerky flow which is parity invariant. It displays chaotic behaviours in two distinct ranges of its control parameter. Deterministic chaos is principally observed from a symmetric limit cycle which after a symmetry-breaking bifurcation gives rise to two cascades of flip bifurcation. Then two coexisting asymmetric chaotic attractors are observed, and after a symmetry-restoring crisis a symmetric chaotic attractor is created. Chaotic attractors also coexist in another very narrow range of control parameter as results of two period doubling cascades of bifurcation from a pair of mutually symmetric coexisting limit cycles. q 2000 Elsevier Science B.V. All rights reserved. PACS: 05.45.Ac; 05.45.-a; 02.30.Hq; 02.60.Cb Keywords: Jerk; Chaos; Attractor; Differential equation; Parity
Recently, some papers w1–5x appeared in response to an exciting question posed by Gottlieb w6x concerning existence of simple jerk equations which may exhibit a chaotic behaviour. A jerk equation is an autonomous third-order differential equation of the form ...
x s j Ž x , x˙ , x¨ .
Ž 1.
Here, x denotes a scalar, for example the position coordinate, the overdot represents the time derivative
)
E-mail: [email protected]
and the jerk function j is the time derivative of the acceleration w7x. The Poincare–Bendixson theorem shows that ´ chaos does not exist in a two-dimensional autonomous system. Since Eq. Ž1. can equivalently be written as three, first-order, ordinary differential equations, a nonlinear jerk function j is a necessary conditions for deterministic chaos. Moreover it is well known that polynomials are the algebraically simplest smooth functions. Therefore, given a general propriety Ž P . for example the type of the nonlinearity: quadratic or cubic etc., we consider the set of the jerk equations which verify Ž P . and have jerk functions which are polynomials with only one alge-
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 8 1 9 - 1
384
J.-M. Malasomar Physics Letters A 264 (2000) 383–389
braic nonlinearity and the minimal number of terms that allow chaotic behaviour. We shall call this set the class of the simplest chaotic jerk equation with the propriety Ž P .. For example let us consider dissipative jerk equations with only quadratic nonlinearities. After an exhaustive numerical investigation of such equations, Sprott found what seems to be the class of the simplest dissipative quadratic jerk equations which exhibit chaotic behaviour w1,2x. This class contains two members, but these equations are equivalent to within a constant. Each equation has a multivariate polynomial jerk function with only three terms including a single quadratic nonlinearity. Symmetry has always played an important role in physics, from fundamental formulations of basic principles to concrete applications, and is present in a variety of chaotic systems. This fact raises a natural question: What is the class of algebraically simplest dissipative jerk equations which are invariant under the parity transformation x yx and that exhibit chaotic behaviour? The problem is to find multivariate polynomial functions j with a single algebraic odd nonlinearity and the minimum number of terms that give dissipative chaotic systems of the form Ž1.. Since cubic nonlinearities are the simplest among odd algebraic nonlinearities we will focus on this case. Examples of such dissipative polynomial jerk equations with only one cubic nonlinearity and chaotic dynamics, are already known. The older of them is the almost forgotten thermal convection model of Moore and Spiegel proposed only three years later than the popular Lorenz model w8x. This system can be written in the nondimensional form
™
...
x q x¨ q Ž T y R q Rx 2 . x˙ q Tx s 0
where R and T are parameters. Solutions to this equation are thought to be chaotic when R s 100 and T s 20 but also in an entire region of the R–T plane w8–10x. Later, Auvergne and Baglin w11x described, in a rough approximation, the motion of an ionization zone in a star by an equation of the same type: ...
x q x¨ q l Ž 1 q m x 2 . x˙ q n x s 0
where the coefficients are given by the following expressions
ls
g1 A2 K 2
,
ms
g2 Ž1 q g1 . g1
2
,
ns
B
1
A A2 K 2
The term A2 K 2 gives a measure of the nonadiabaticity and the term ArB can be considered as a source term of the star. Chaotic behaviour occurs for example with g 1 s 0.3, g 2 s 25.0, A2 K 2 s 0.01 and BrA s 0.12 or BrA s 0.022. It is also the case of the Arneodo–Coullet–Spie´ gel–Tresser polynomial jerk equation ...
x q Ax¨ q Bx˙ q Cx q Dx 3 s 0
which exhibits chaotic dynamics for various values of the four parameters w12–14x. We can use for example the following sets A s 1, B s 3.5,C s 9.6, D s y14 or A s 1, B s 3.5,C s y5.5, D s 14 . These examples demonstrate the existence of polynomial jerk functions with four terms including a single cubic nonlinearity, for which Eq. Ž1. has chaotic solutions. In a more recent paper w1x, Sprott reported the results of an almost systematic numerical examination of third-order, one-dimensional, autonomous, ordinary differential equations with cubic nonlinearities and chaotic dynamics. Four functionally distinct forms, not given in his paper, were found with four terms and one cubic nonlinearity. But surprisingly no cases as simple as the quadratic nonlinearity, i.e. with three terms including a single nonlinearity, were found by this method. The aim of this Letter is to give such an example and also to analyze with some details its chaotic dynamics. The system to be considered is the following ...
x q a x¨ y xx˙ 2 q x s 0
Ž 2.
where the parameter a is restricted to be positive in order to have a dissipative flow. Eq. Ž2. can be derived from the Newtonian equation: x¨ s ya x˙ q
t
Ht x Ž t .
x˙ 2 Ž t . y 1 dt
0
governing the one-dimensional motion of point particle of mass unit under the influence of a force that depends on the instantaneous velocity Žviscous
J.-M. Malasomar Physics Letters A 264 (2000) 383–389
damping. but also on the position and velocity history. Eq. Ž2. has a single fixed point at the origin of the phase space which eigenvalues satisfy the characteristic equation P Ž l. s l3 q al2 q 1 s 0. An elementary study proves that this polynomial has only one real root lr - y2 ar3 which is therefore negative. Since the characteristic equation is a cubic equation with real coefficients, we will have without loss of generality P Ž l. s Ž l y l r .Ž l y l c .Ž l y lc . where lc is a complex number. After expanding the above equation and comparing the coefficients with those of the original characteristic equation we come up with the relation ReŽ l c . s 2y1ly2 r . We conclude that the real part of lc is positive and that the fixed point is unstable. The negative real eigenvalue is associated with a one-dimensional stable manifold, whereas both complex conjugate eigenvalues, with positive real parts, are associated with a two-dimensional unstable manifold in which trajectories are spiraling outwards. Thus the origin of the phase space is a saddle-focus with an instability index of 2. We note that Eq. Ž2. is invariant under the parity transformation
Ž x , y, z .
385
For a ) 2.540 all solutions numerically found are unbounded. At the contrary for a Q 2.540, depending on initial conditions, solutions of Eq. Ž2. are either unbounded or they settle down on a symmetric stable limit cycle. This stable solution has been studied via a Poincare´ section S defined by
S s Ž x , z . g R 2 N y s 0, y˙ ) 0 4
Ž 3.
Its basin of attraction appears to be very small and Fig. 1 shows its trace with the plane S for a s 2.5. We can notice that this section is obviously parity invariant. The basin exhibits also three spiraling structures around the fixed point at origin of the phase space and around the two symmetric points Ž"3.81,0,. 25.08.. For the same value a s 2.5 simple initial conditions which belong to this basin are given for example by Ž"4,0,0.. When the control parameter reaches the value a s 2.196 . . . , this limit cycle loses stability in a symmetry-breaking bifurcation. Therefore, it becomes unstable and two coexisting asymmetric stable limit cycles are created. One is symmetric of the other. When the parameter a is further decreased
™ Ž yx ,y y,y z .
of the phase space R 3 s Ž x, y s x, ˙ z s x¨ .4. Therefore a solution of Eq. Ž2. that is invariant under this transformation is called a symmetric solution, whereas a solution that is not invariant under the same transformation is called an asymmetric solution. Consequently, for a specified value of the control parameter a , if an asymmetric solution is found, an other different asymmetric solution can be obtained by applying the above transformation. Hence, all asymmetric solutions occur in pairs. Scanning the range 0 - a - 5, in increments of 10y2 , we have analyzed numerical solutions for 10 6 randomly chosen initial conditions with N x N- 5, N x˙ N- 5, and N x¨ N- 5 for each value of a . The calculations were performed using a fourth-order Runge–Kutta algorithm with step size of D t s 10y3 . Since the most common dynamics involve unbounded solutions, we stop the numerical calculations whenever we detect the condition N x N qN x˙ N qN x¨ N) 10 5. When bounded solutions were found, the results were verified by using a 7–8 Runge– Kutta–Fehlberg adaptive step algorithm.
Fig. 1. Intersection with the plane x˙ s 0 of the basin of attraction of the symmetric stable limit cycle for the control parameter value a s 2.5. This section is clearly parity invariant. Three spiraling structures are clearly visible.
386
J.-M. Malasomar Physics Letters A 264 (2000) 383–389
both limit cycles lose together their stability through period doubling bifurcation at a s 2.105 . . . It is the beginning of two complete cascades of bifurcation that may be observed depending on the initial conditions and that lead to chaotic dynamics as illustrated by the bifurcation diagram shown in Fig. 2. In this figure, successive local minima of x, computed using Ž3. by the classical Henon trick w15x, ´ are recorded after transients have decayed, and plotted versus a . As a is decreased both attractors enlarge within their own basin of attraction. When the control parameter reaches the value a s 2.0644 . . . , they both simultaneously experience a tangency with the common basin boundary separating their two basins and a symmetry restoring crisis occurs. After the crisis, the asymptotic motion settles Fig. 3. Variations of the largest, non zero, Lyapunov exponent l m versus the parameter a in the same range as in Fig. 2. The onset of chaos is visible near a , 2.084, where l m first becomes positive.
down to a larger symmetric chaotic attractor. With further decrease of a this attractor enlarge within its basin of attraction and collides with the basin boundary at a s 2.0278 . . . It is the final boundary crisis which kills the symmetric chaotic attractor. The chaotic region falls in the range 2.0278 . . . - a 2.0840 . . . which is extraordinary tiny, and this is probably why this flow was missed during the Sprott’s earlier numerical search w1x. It is worth noticing that these two bifurcation diagrams are very similar to those obtain for a cubic map of the interval. For example let us consider the map fmŽ x . s ym x y x 3. The fixed point x s 0 is stable in the interval x y 1,1w and when m - y1 a pair of mutually symmetric equilibrium points x "s " y 1 y m which are stable in the domain x y 2,y 1w exist. At m s y1 two cascades of period doubling bifurcation develop when the parameter m decreases, cumulating in two asymmetric chaotic attractors. Then a symmetry-restoring crisis occurs at m s y2.599 and a symmetric chaotic attractor is created which is destroyed in a boundary crisis at m s y3. Traditionally, the Lyapunov exponents are used to distinguish a periodic solution from a chaotic behaviour. They have numerically been determined by
'
Fig. 2. Bifurcation diagram of successive local minima x min of x in the Poincare´ section Ž3. plotted versus control parameter a . The end of the cascade of period doubling bifurcation leading to chaos is clearly visible for both the coexisting asymmetric periodic solutions Žtop and bottom. when a decreases.
J.-M. Malasomar Physics Letters A 264 (2000) 383–389
a standard numerical method w16x for a in the same range as in Fig. 2. Over the intervals in which the system has stable periodic solutions, the Lyapunov exponents spectra takes the form Ž l1 , l2 , l3 . s Ž0,y,y ., while over the intervals in which the system exhibits chaos, the Lyapunov exponents spectra takes the form Ž l1 , l2 , l3 . s Žq,0,y .. Fig. 3 shows the variations of the largest, non zero, exponent l m versus parameter a . The onset of chaos is visible near a s 2.0840 . . . , where l m first becomes positive. For a - 2.0840 . . . the exponent l m generally increases, except for the dips caused by the windows of stable periodic behaviour. The most visible are the large dips due to the period-3 window near a , 2.043 and due to the period-5 windows near a , 2.0545 for the symmetric attractor. Also we can notice the period-3 window near a , 2.0743 for both asymmetric attractors. Fig. 4 shows a phase space portrait of the symmetric chaotic attractor for a s 2.05. The Lyapunov exponents are found to be in base-e: l1 s 0.0541, l2 s 0, and l3 s y2.1041. Therefore the corresponding Kaplan–Yorke dimension w17x is D KY s 2 y l1rl3 , 2.0257. Using the Poincare´ section S , a first return map is hereafter computed and displayed in Fig. 5 for a s 2.05. To a very good approximation all points appear to lie on a single curve and a cubic map constituted by three monotonic branches separated by two critical points given by C1 s y3.469 . . . and C2 s y3.088 . . . is obtained. The time series of x and its first three time-de... rivatives y s x, ˙ z s x¨ and j s x for a typical trajectory onto the attractor at a s 2.05 are displayed in Fig. 6. The average duration between two successive intersections of the trajectory with the Poincare´ sec-
Fig. 4. Phase space portrait of the symmetric chaotic attractor for the parameter value a s 2.05.
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Fig. 5. First return map to the Poincare´ section S at a s 2.05. The map appears to be one-dimensional constituted by three monotonic branches separated by two critical points given by C1 sy3.469 . . . and C2 sy3.088 . . . .
tion S is found numerically to be T , 15.2. Therefore the contraction of the phase space per cycle is given by eTl 3 , 10y1 2 which helps to explain why the first return map of Fig. 5 looks like a one-dimensional map. We have also found another interval of the control parameter in which chaotic behaviour is present. For 0.0018 Q a , depending on initial conditions, the solutions of Eq. Ž2. are either unbounded or they settle down on two mutually symmetric coexisting stable
Fig. 6. Time series of x and its first three time-derivatives y s x, ˙ ... z s x¨ and js x for a typical trajectory onto the attractor displayed in Fig. 4 for a s 2.05.
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limit cycles. These solutions have been studied again via the Poincare´ section Ž3.. Their basins of attraction are extraordinary tiny with a very complex structure which is still under investigation. For example, simple initial conditions within these basins are Ž"5.0," 0.926," 0.15. when a s 0.07. When a is increased these cycles remain stable until a s 0.0724 . . . . Then they simultaneously lose their stability in two cascades of period doubling bifurcation leading to coexisting chaotic attractors. Beyond these cascades to chaos, the attractors develop as a is raised. After many band-merging bifurcations each diagram is constituted by four remaining bands. For one of these diagrams, these bands have separately been plotted in Fig. 7 in order to enlarge the vertical scale. Two interior crises w18x are visible which causes the sudden expansion of the chaotic bands at a , 0.075355 and at a , 0.07536. Fig. 8 shows the variations of the largest, non zero, exponent l m versus parameter a . The onset of chaos is visible near a s 0.0753514 . . . , where l m first becomes positive. The exponent l m globally increases until the final boundary crisis. Once again,
Fig. 8. Variations of the largest, non zero, Lyapunov exponent l m versus the parameter a in the same range as in Fig. 7. The onset of chaos is visible near a , 0.0753514, where l m first becomes positive.
it is worth noticing that these coexisting chaotic dynamics fall in the extraordinary small range 0.0753514 . . . - a - 0.0753624 . . . which is five
Fig. 7. Bifurcation diagram of successive local minima x min of x in the Poincare´ section Ž3. plotted versus control parameter a . Since after many band-merging bifurcations, the diagram is constituted by four bands, they have separately been plotted in order two enlarge the vertical scale.
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thousand times smaller than the case previously reported. In conclusion, we have shown that deterministic chaos exists in a polynomial dissipative jerk equation whose jerk function is constituted by only three terms including a single cubic nonlinearity. Moreover, the flow associated with this equation is parity invariant. Two different parameter ranges where a such behaviour exists have been found. The route to chaos is the standard cascade of period doubling. In both intervals we have observed locally coexisting chaotic attractors. Any polynomial jerk function with only three terms depends on three coefficients. By rescaling x and t it is always possible to normalize two of them. Therefore only one control parameter remains free. Our equation has the minimum number of terms that allows an adjustable control parameter and has only one cubic nonlinearity. Therefore it is unlikely that any algebraically simpler dissipative cubic polynomial form of jerky chaotic flows exists. We conjecture that Eq. Ž2. belongs to the class of the simplest dissipative chaotic jerk equations which are parity invariant. However it is difficult to rule out the possibility of finding new equations which belong to the same class.
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References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x
w18x
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