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A hypothesis concerning ‘quantal’ Hilbert space criterion of chaos in nonlinear dynamical systems
Chaos,
Pergamon
Solironr
& FracraLs Vol. 7, No. 5, pp. 653-657, 19% Copyright @ 19% Elsevier Science Ltd Printed in Great Britain. All rights resewed o%o-0779/96 $15.00 + 0.00
0960-0779(94)00213-4
A Hypothesis
Concerning ‘Quantal’ Hilbert Space Criterion Chaos in Nonlinear Dynamical Systems KRZYSZTOF
Department
of Theoretical
Physics,
University
of
KOWALSKI of t6dt,
ul. Pomorska
149/153,
90-236
K6di,
Abstract-Based on the Hilbert space approach to the theory of nonlinear dynamical developed by the author, a hypothesis is formulated concerning the ‘quantal’ criterion for ordinary differential systems to exhibit chaotic behaviour. Copyright @ 1996 Elsevier Science
Poland
systems classical Ltd.
1. INTRODUCTION
As so many natural laws and models of natural phenomena are described by nonlinear dynamical systems dx/dt = F(x), it is no exaggeration to say that they abound in modern science and technology. Physical and numerical experiments show that deterministic chaos in such systems is ubiquitous. On the other hand, the recent observations concerning the possibility of controlling chaos [l] indicate its new promising applications in engineering and medical sciences [2]. No wonder the interest in the theory of chaotic dynamical systems is steadily increasing. The object of the present paper is to formulate a hypothesis concerning ‘quantal’ Hilbert space criterion of chaos in nonlinear systems dx/dt = F(x), where F is analytic in x, with the use of the formalism developed by the author [3-91, relying on reduction of nonlinear dynamical systems to the linear, abstract, Schrodinger-like equation in Hilbert space. Namely, it is suggested that the chaotic behaviour of the nonlinear dynamical systems can be related to the growth of the ‘quantal’ entropy which can be naturally introduced within the Hilbert space approach resembling quantum mechanics.
2. HILBERT
SPACE APPROACH
In this section we briefly outline the Hilbert space description of nonlinear ordinary differential systems [3]. Consider the analytic system (complex or real) dz = F(z), dt
40) = zo,
where F: Ck + Ck is analytic in z. The vectors )z , t ) defined by
where lz) is a normalized coherent state (see Appendix) 653
and z fulfils (l),
satisfy the
654
K. KOWALSKl
following linear, Schrodinger-like
equation in Hilbert
“f’Z> r) = M(z. t),
space: lz, 01 = iZ(J>
(3)
where M is the boson operator such that M = a:’ - F(a).
(4)
In view of (2) we arrive at the following eigenvalue equation: ah,. t> = Zh”, Olzo, fj
(3
relating the solution z(za, t) of (1) and the solution /zO, t) of (3). Thus, it turns out that the nonlinear dynamical system (1) can be cast into the linear, abstract, Schrodinger-like equation (3). We note that the algorithm can be immediately extended to the case of nonautonomous systems such that the corresponding vector field is analytic in z-variables. We also remark that the eigenvalue equation (5) suggests the ‘quantization scheme’ of the form z -+ a, where z fulfils (l), within introduced formahsm resembling quantum mechanics. Observe that (3) and (5) correspond to the ‘Schrodinger picture’ within the actual ‘quantal’ treatment. We end this brief account of the Hilbert space approach with discussion of the ‘Heisenberg picture’. The ‘Heisenberg equations of motion’ obeyed by the time-dependent Bose annihilation operators are of the form -$-
= [a, Ml:
a(O) = a,
(6)
where M is the ‘Hamiltonian’ given by (4). Since the ‘Hamiltonian’ in the ‘Schriidinger picture’ N is independent of time, it therefore coincides with the ‘Hamiltonian’ in the ‘Heisenberg picture’. The formal solution of (6) can be written as - V(t)-‘aV(t),
a(t)
where V(t) = eNM is the evolution operator. It is clear that the following relation holds true:
ij -
Iz(,-
v(t)l~,>,
where jza, t) is the solution of (3). Now equations (5), (8) and (7) taken together yield a(tjl2,))
=
z(z,,,
t)l~d.
Hence we find that the solutions of the systems (1) coincide with the expectation (covariant symbols) of the time-dependent Bose annihilation operators, i.e. zh.
t) - (zoia(t)lzo>.
(9)
values (10)
It should be noted that whenever the ‘quantization scheme’ z --+ a is assumed, where z fulfils (l), then (10) forms the ‘Ehrenfest’s theorem’ within the actual ‘quantal’ approach.
3. THE CRITERION
OF CHAOS
We now formulate the hypothesis concerning the criterion for the nonlinear dynamical systems (1) to show chaotic behaviour. Consider the system (1). We introduce the operator
Chaos in nonlinear dynamical systems
655
p of the form (see (A2)): P=
Evidently,
I RZk
the operator p is Hermitian
Furthermore,
d&w)e-IW-Z0~21w) (WI. and nonnegative,
using (A6) we immediately
i.e.
p+= P,
(124
pa 0.
Wb)
obtain Trp = 1.
Finally,
taking into account (ll),
(11)
WC)
(A2), (A5) and (10) we get (a(t))
= Tr(pa(t))
= z(t),
(124
where a(t) are the time-dependent Bose annihilation operators and z(t) fulfils (1). We have thus shown that if the ‘quantization scheme’ z + a, where z satisfies (1) is assumed, then p plays the role of the density matrix within the ‘quantal’ Hilbert space approach. Prompted by this analogy, we define the ‘entropy’ as S = -Tr(plnp).
(13)
On using the relation 1 = -, n 3 1, 2” - 1 following directly from (11)) (A2) and (A4), we find Trp”
S = -iLlnl=
i42’
2’
(14)
21n2.
(15)
Since an arbitrary vector in a Fock space can be specified by an infinite sequence of i-vectors of 3.‘~ and O’s and the number of different i-vectors is 2’, therefore the obtained value of the ‘entropy’ can be regarded as an averaged amount of the information necessary to fix the coherent state Iz,,) in a Hilbert space of states. Now in analogy to quantum mechanics, we introduce the time-dependent ‘density matrix’ such that p(t) = lRzkdp(w)e-IW-Z012efMIw) (wle“@,
where M is the ‘Hamiltonian’ related to the system (1). We note that p(t) is Hermitian and nonnegative at any time. ‘entropy’ corresponding to (16) is given by S(t) = -Tr[p(t)lnp(t)]. We are now in a position conditions hold:
to propound
our hypothesis.
3 C-1, v (z(zo, t)12
f*70
WI
The time-dependent (17)
We assert that if the following
(184
K. KOWALSKI
6%
where z(q,, t) is the solution to (1); C :> 0 is a constant.
\d _dStr) > (j . -i- :, dt then the system (1) is a chaotic one. 4. DISCUSSION
We have formulated in this work a hypothesis concerning a new, ‘quantal Hilbert space criterion of chaos in nonlinear dynamical systems. Note that the condition (Ba) ensures that the ‘density matrix’ p(t) is of trace class for all t 12 t*. Indeed, from (16) and the relation (zo, tjzo. t) - exp(lz(z,,, t)l” - izJ’). (19) which is an immediate consequence of (2). it follows that Trp(t)
= ^ d&v)expC-lw - qj12)exp(!z(w, t)]” - /w12), (20) i R’” where z(z+ t) is the soiution to (1). On the other hand, the condition (%a) is consistent with the exceptional role of dissipativity of systems showing chaotic behaviour. We remark that the nonunitary evolution implied by the Schrijdinger-like equation (3) is crucial for the actual treatment. In fact, as in quantum mechanics: the unitary evolution implies s(t) = S = const. We note at the same time that the evolution operator V(t) = etM defined by (8) is unitary only in the case of the linear system (1) with a skew-Hermitian fundamental matrix. Finally, we point out that the ‘quantal’ Hilbert space approach recognizes the divergence of orbits in the phase space of the system (1). Namely. we have the formula -t bz,,, t)? __._- ._._-__j(z(), tlz”----L---= exp ( -iz(zO + (32(,, t) - z(zg, t)l’) (21) (Z(), t!Zg> t)(zfJ + hz,,, tjz,, + h,,, r> which can be easily obtained from (2) and (Al). Therefore, the growth of the distance between the two trajectories of the system (1) &art@ from z. and z. -!- &Q implies the decay of the corresponding transition probability from the left-hand side of (21).
REFERENCES CY. Grebogi and J. A. Yorke, P/zys. Rev. Lett. 64, 11% (1990). Garfinkel, M. L. Spano, W. L. Ditto and J. N. Weiss, Science 257. 1230 (1992). Kowalski, Phy.&a A 145, 408 (1987). Kowalski, Physica A 152, 98 (1988). Kowalski and W.-H. Steeb. Progr. Theor. f’hys. 85, 713 (1991). K. KowalskiandW.-H. Steeb,Progr. Theor. Phys. 89,975 (1991). K. Kowalski and W.-H. Steeb. Nonlinear Dvnamical Systems and Carleman Linearization. Singapore (1991 j. K. Kowalski, Physica A 195, 137 (1993). K. Kowalski, Physica A 198, 493 (1993). F. A. Berezin. Mat. Sh. 86, 578 (1971) (Russian).
i. E. Ott,
2. 3 4 5. 6.
7, X. 9. 10.
A. K. K. K.
World Scientific,
APPENDIX: COHERENT STATES We append the basic facts about coherent states. Recall first that the Bose creation (a’) and annihilation operators. where a+ = (0:. a:‘). a = (CT. ._a;). obey the Heisenberg algebra
(a)
Chaos in nonlinear dynamical systems
[Uj,q = &jZ, [a,,Uj]= [cg,u;] = 0,
i, j = 1,
. ., k.
The coherent states lz), where z E C k, are usually defined as eigenvectors of the annihilation alz)
657
operators, that is
= ~12).
The normalized coherent states can be defined as Iz) = exp (- ilzl’) exp (z * a+)lO). where lz12 = Ct,lzil’,
u ‘v = ~giz+u,
and IO) is the vacuum vector satisfying alO) = 0.
The coherent states are not orthogonal, namely (zlw) = exp(- #J/’ + lw12 - 22*.w)], where the asterisk designates the complex conjugation and z* = (t?, ., zt). Hence we find l(zlw)12 = exp(-lz -’ ~1~).
(AU set. The resolution of the identity for the coherent states is
These states form the complete (overcomplete) given by
where k 1
44(z) = PI- d(Rezd d(Imz,). Now let I@) be an arbitrary state. It can be easily shown that the function (symbol of the vector) @(z*) = (z\$) is of the form Hz*) = &z*)exp(-ilz12), 643) where &z*) is an analytic (entire) function. Taking into account (A2) and (A3) we get
ddz)exp(- lzl’Nkz*))*&*). (@IV)= IRZk Thus the abstract vectors can be represented by analytic functions. This representation is usually known as the Bergmann representation. The action of the Bose operators in the Bergmann representation has the following form: a&z*)
= $&z*),
at&z*)
= z*&z*).
On using (A2) and (A3) we arrive at the following reproducing property of coherent states: kw*,
= JaZkdy(z)exp(-l~l~)Sl(w*, Z&Z*),
(A41
where the reproducing kernel (Bergman reproducing kernel) is X(w*, z) = exp(w*.z). Taking the Hermitian conjugate of (A4) we obtain the following form of the reproducing property: d&)exp(-lz12)exp(w
.z*)$(z).
(-45)
We finally remark that the coherent states are the convenient tool for the study of operators. For example, the trace of a linear operator L can be expressed by TrL
=
I I?
dp(z)L(z”,
where L(z*,
is called the covariant symbol of the operator L [IO].
z) = (ZlLlZ)
4,
(‘46)
Pergamon
Solironr
& FracraLs Vol. 7, No. 5, pp. 653-657, 19% Copyright @ 19% Elsevier Science Ltd Printed in Great Britain. All rights resewed o%o-0779/96 $15.00 + 0.00
0960-0779(94)00213-4
A Hypothesis
Concerning ‘Quantal’ Hilbert Space Criterion Chaos in Nonlinear Dynamical Systems KRZYSZTOF
Department
of Theoretical
Physics,
University
of
KOWALSKI of t6dt,
ul. Pomorska
149/153,
90-236
K6di,
Abstract-Based on the Hilbert space approach to the theory of nonlinear dynamical developed by the author, a hypothesis is formulated concerning the ‘quantal’ criterion for ordinary differential systems to exhibit chaotic behaviour. Copyright @ 1996 Elsevier Science
Poland
systems classical Ltd.
1. INTRODUCTION
As so many natural laws and models of natural phenomena are described by nonlinear dynamical systems dx/dt = F(x), it is no exaggeration to say that they abound in modern science and technology. Physical and numerical experiments show that deterministic chaos in such systems is ubiquitous. On the other hand, the recent observations concerning the possibility of controlling chaos [l] indicate its new promising applications in engineering and medical sciences [2]. No wonder the interest in the theory of chaotic dynamical systems is steadily increasing. The object of the present paper is to formulate a hypothesis concerning ‘quantal’ Hilbert space criterion of chaos in nonlinear systems dx/dt = F(x), where F is analytic in x, with the use of the formalism developed by the author [3-91, relying on reduction of nonlinear dynamical systems to the linear, abstract, Schrodinger-like equation in Hilbert space. Namely, it is suggested that the chaotic behaviour of the nonlinear dynamical systems can be related to the growth of the ‘quantal’ entropy which can be naturally introduced within the Hilbert space approach resembling quantum mechanics.
2. HILBERT
SPACE APPROACH
In this section we briefly outline the Hilbert space description of nonlinear ordinary differential systems [3]. Consider the analytic system (complex or real) dz = F(z), dt
40) = zo,
where F: Ck + Ck is analytic in z. The vectors )z , t ) defined by
where lz) is a normalized coherent state (see Appendix) 653
and z fulfils (l),
satisfy the
654
K. KOWALSKl
following linear, Schrodinger-like
equation in Hilbert
“f’Z> r) = M(z. t),
space: lz, 01 = iZ(J>
(3)
where M is the boson operator such that M = a:’ - F(a).
(4)
In view of (2) we arrive at the following eigenvalue equation: ah,. t> = Zh”, Olzo, fj
(3
relating the solution z(za, t) of (1) and the solution /zO, t) of (3). Thus, it turns out that the nonlinear dynamical system (1) can be cast into the linear, abstract, Schrodinger-like equation (3). We note that the algorithm can be immediately extended to the case of nonautonomous systems such that the corresponding vector field is analytic in z-variables. We also remark that the eigenvalue equation (5) suggests the ‘quantization scheme’ of the form z -+ a, where z fulfils (l), within introduced formahsm resembling quantum mechanics. Observe that (3) and (5) correspond to the ‘Schrodinger picture’ within the actual ‘quantal’ treatment. We end this brief account of the Hilbert space approach with discussion of the ‘Heisenberg picture’. The ‘Heisenberg equations of motion’ obeyed by the time-dependent Bose annihilation operators are of the form -$-
= [a, Ml:
a(O) = a,
(6)
where M is the ‘Hamiltonian’ given by (4). Since the ‘Hamiltonian’ in the ‘Schriidinger picture’ N is independent of time, it therefore coincides with the ‘Hamiltonian’ in the ‘Heisenberg picture’. The formal solution of (6) can be written as - V(t)-‘aV(t),
a(t)
where V(t) = eNM is the evolution operator. It is clear that the following relation holds true:
ij -
Iz(,-
v(t)l~,>,
where jza, t) is the solution of (3). Now equations (5), (8) and (7) taken together yield a(tjl2,))
=
z(z,,,
t)l~d.
Hence we find that the solutions of the systems (1) coincide with the expectation (covariant symbols) of the time-dependent Bose annihilation operators, i.e. zh.
t) - (zoia(t)lzo>.
(9)
values (10)
It should be noted that whenever the ‘quantization scheme’ z --+ a is assumed, where z fulfils (l), then (10) forms the ‘Ehrenfest’s theorem’ within the actual ‘quantal’ approach.
3. THE CRITERION
OF CHAOS
We now formulate the hypothesis concerning the criterion for the nonlinear dynamical systems (1) to show chaotic behaviour. Consider the system (1). We introduce the operator
Chaos in nonlinear dynamical systems
655
p of the form (see (A2)): P=
Evidently,
I RZk
the operator p is Hermitian
Furthermore,
d&w)e-IW-Z0~21w) (WI. and nonnegative,
using (A6) we immediately
i.e.
p+= P,
(124
pa 0.
Wb)
obtain Trp = 1.
Finally,
taking into account (ll),
(11)
WC)
(A2), (A5) and (10) we get (a(t))
= Tr(pa(t))
= z(t),
(124
where a(t) are the time-dependent Bose annihilation operators and z(t) fulfils (1). We have thus shown that if the ‘quantization scheme’ z + a, where z satisfies (1) is assumed, then p plays the role of the density matrix within the ‘quantal’ Hilbert space approach. Prompted by this analogy, we define the ‘entropy’ as S = -Tr(plnp).
(13)
On using the relation 1 = -, n 3 1, 2” - 1 following directly from (11)) (A2) and (A4), we find Trp”
S = -iLlnl=
i42’
2’
(14)
21n2.
(15)
Since an arbitrary vector in a Fock space can be specified by an infinite sequence of i-vectors of 3.‘~ and O’s and the number of different i-vectors is 2’, therefore the obtained value of the ‘entropy’ can be regarded as an averaged amount of the information necessary to fix the coherent state Iz,,) in a Hilbert space of states. Now in analogy to quantum mechanics, we introduce the time-dependent ‘density matrix’ such that p(t) = lRzkdp(w)e-IW-Z012efMIw) (wle“@,
where M is the ‘Hamiltonian’ related to the system (1). We note that p(t) is Hermitian and nonnegative at any time. ‘entropy’ corresponding to (16) is given by S(t) = -Tr[p(t)lnp(t)]. We are now in a position conditions hold:
to propound
our hypothesis.
3 C-1, v (z(zo, t)12
f*70
WI
The time-dependent (17)
We assert that if the following
(184
K. KOWALSKI
6%
where z(q,, t) is the solution to (1); C :> 0 is a constant.
\d _dStr) > (j . -i- :, dt then the system (1) is a chaotic one. 4. DISCUSSION
We have formulated in this work a hypothesis concerning a new, ‘quantal Hilbert space criterion of chaos in nonlinear dynamical systems. Note that the condition (Ba) ensures that the ‘density matrix’ p(t) is of trace class for all t 12 t*. Indeed, from (16) and the relation (zo, tjzo. t) - exp(lz(z,,, t)l” - izJ’). (19) which is an immediate consequence of (2). it follows that Trp(t)
= ^ d&v)expC-lw - qj12)exp(!z(w, t)]” - /w12), (20) i R’” where z(z+ t) is the soiution to (1). On the other hand, the condition (%a) is consistent with the exceptional role of dissipativity of systems showing chaotic behaviour. We remark that the nonunitary evolution implied by the Schrijdinger-like equation (3) is crucial for the actual treatment. In fact, as in quantum mechanics: the unitary evolution implies s(t) = S = const. We note at the same time that the evolution operator V(t) = etM defined by (8) is unitary only in the case of the linear system (1) with a skew-Hermitian fundamental matrix. Finally, we point out that the ‘quantal’ Hilbert space approach recognizes the divergence of orbits in the phase space of the system (1). Namely. we have the formula -t bz,,, t)? __._- ._._-__j(z(), tlz”----L---= exp ( -iz(zO + (32(,, t) - z(zg, t)l’) (21) (Z(), t!Zg> t)(zfJ + hz,,, tjz,, + h,,, r> which can be easily obtained from (2) and (Al). Therefore, the growth of the distance between the two trajectories of the system (1) &art@ from z. and z. -!- &Q implies the decay of the corresponding transition probability from the left-hand side of (21).
REFERENCES CY. Grebogi and J. A. Yorke, P/zys. Rev. Lett. 64, 11% (1990). Garfinkel, M. L. Spano, W. L. Ditto and J. N. Weiss, Science 257. 1230 (1992). Kowalski, Phy.&a A 145, 408 (1987). Kowalski, Physica A 152, 98 (1988). Kowalski and W.-H. Steeb. Progr. Theor. f’hys. 85, 713 (1991). K. KowalskiandW.-H. Steeb,Progr. Theor. Phys. 89,975 (1991). K. Kowalski and W.-H. Steeb. Nonlinear Dvnamical Systems and Carleman Linearization. Singapore (1991 j. K. Kowalski, Physica A 195, 137 (1993). K. Kowalski, Physica A 198, 493 (1993). F. A. Berezin. Mat. Sh. 86, 578 (1971) (Russian).
i. E. Ott,
2. 3 4 5. 6.
7, X. 9. 10.
A. K. K. K.
World Scientific,
APPENDIX: COHERENT STATES We append the basic facts about coherent states. Recall first that the Bose creation (a’) and annihilation operators. where a+ = (0:. a:‘). a = (CT. ._a;). obey the Heisenberg algebra
(a)
Chaos in nonlinear dynamical systems
[Uj,q = &jZ, [a,,Uj]= [cg,u;] = 0,
i, j = 1,
. ., k.
The coherent states lz), where z E C k, are usually defined as eigenvectors of the annihilation alz)
657
operators, that is
= ~12).
The normalized coherent states can be defined as Iz) = exp (- ilzl’) exp (z * a+)lO). where lz12 = Ct,lzil’,
u ‘v = ~giz+u,
and IO) is the vacuum vector satisfying alO) = 0.
The coherent states are not orthogonal, namely (zlw) = exp(- #J/’ + lw12 - 22*.w)], where the asterisk designates the complex conjugation and z* = (t?, ., zt). Hence we find l(zlw)12 = exp(-lz -’ ~1~).
(AU set. The resolution of the identity for the coherent states is
These states form the complete (overcomplete) given by
where k 1
44(z) = PI- d(Rezd d(Imz,). Now let I@) be an arbitrary state. It can be easily shown that the function (symbol of the vector) @(z*) = (z\$) is of the form Hz*) = &z*)exp(-ilz12), 643) where &z*) is an analytic (entire) function. Taking into account (A2) and (A3) we get
ddz)exp(- lzl’Nkz*))*&*). (@IV)= IRZk Thus the abstract vectors can be represented by analytic functions. This representation is usually known as the Bergmann representation. The action of the Bose operators in the Bergmann representation has the following form: a&z*)
= $&z*),
at&z*)
= z*&z*).
On using (A2) and (A3) we arrive at the following reproducing property of coherent states: kw*,
= JaZkdy(z)exp(-l~l~)Sl(w*, Z&Z*),
(A41
where the reproducing kernel (Bergman reproducing kernel) is X(w*, z) = exp(w*.z). Taking the Hermitian conjugate of (A4) we obtain the following form of the reproducing property: d&)exp(-lz12)exp(w
.z*)$(z).
(-45)
We finally remark that the coherent states are the convenient tool for the study of operators. For example, the trace of a linear operator L can be expressed by TrL
=
I I?
dp(z)L(z”,
where L(z*,
is called the covariant symbol of the operator L [IO].
z) = (ZlLlZ)
4,
(‘46)