The structural stability of a mathematical model with respect to small changes is a necessary condition for its correctness. The same condition is also necessary for the applicability of numerical methods, a computational experiment. But after S. Smale's works it became clear that in smooth dynamics the system of a general form is not structurally stable, therefore there is no strict mathematical basis for modeling and computational analysis of systems. The contradiction appeared in science: according to physicists dynamics is simple and universal. The paper proposes a solution to this problem based on the construction of dynamic quantum models (DQM). DQM is a perturbation of a smooth dynamical system by a Markov cascade (time is discrete). The dynamics obtained in this way are simpler than smooth dynamics: the structurally stable DQM realizations are everywhere dense and open on the set of all DQM realizations. This dynamics in contrast to the classical one has a clear structural theory, which makes it possible to construct effective algorithms for study of concrete systems. For example this paper shows the use of computer simulation for rigorous proof of hyperbolicity of the Henon system attractor. On the other hand, when fluctuations tend to zero, i.e. in the semiclassical limit, the dynamics of the DQM goes into the initial smooth dynamics. In this paper the equivalence of structural stability and hyperbolicity for smooth discrete dynamical systems is established along this path.
Computational modeling derives from two steps: (i) modeling, i.e. finding a model description of a real system, and (ii) solving the resulting model equations using computational methods [1]. Computational modeling has been used in physics, chemistry and related engineering for many decades because in practice hardly any model equations of systems of interest can be solved analytically, and this is where the computer comes in [2].
However, if an arbitrarily small perturbation of the model leads to a qualitatively different picture of the dynamics, then such a model is not applicable to the real process: strictly speaking, perturbations are included in the definition of a model. Therefore traditionally the stability of a mathematical model with respect to relatively small changes is a necessary condition for its correctness [3]. The same stability condition is necessary for applicability of numerical methods, computational experiments since they inevitably lead to errors of discretization and rounding in calculations [4].
The qualitative invariance of a mathematical model under small perturbations is usually called structural stability. This formally means equivalence, in some exact sense, between the model and its small enough perturbation. For the smooth dynamical systems (sets of differential or difference equations) this equivalence is usually a homeomorphism between the phase portraits of these systems. Such theory of a structural stability going back to H. Poincare, has been developed by A.Andronov and L. Pontrjagin in the case of small dimension of the phase space (1 or 2) [5]. However, the optimism generated by the successes of this theory disappeared after S. Smale's works [6]. It was shown in [7] that when the phase space has larger dimension, then there exist smooth dynamic systems whose neighborhoods do not contain any structurally stable system. For the theory of smooth dynamical systems (its old name is the qualitative theory of differential equations) this result has the same value as Liouville’s theorem on insolvability of the differential equations in quadratures has for the theory of their integration. Namely, it shows that the problem of full topological classification of smooth dynamical systems is hopeless. This means also that there is no strict mathematical basis for modeling and computational analysis. The contradiction has appeared in science, because physicists believe that the dynamics is simple and universal [8].
The paper proposes solution to this problem, based on the construction of dynamic quantum models (DQM). It turns out that taking into account random fluctuations, necessary for the transition to the quantum model of reality, allows us to return in fact to the simple picture of A. Poincare’s dynamics: a dense set of structurally stable systems.
DQM is so named because for Hamiltonian systems it is simply related to the corresponding Schrödinger equation, and its construction is the basis of the method for solving spectral problems [9]. But the definition of DQM is not formally related to Hamiltonian systems; it is defined for any ordinary differential equation or any diffeomorphism on any smooth Riemannian manifold.
The structural stability of the general form DQM opens the way to a mathematically grounded numerical analysis of the dynamics. As an example, this paper shows the use of computer simulation for rigorous proof of hyperbolicity of the Henon system attractor [10] at certain values of parameters. DQM is the natural basis for solving the traditional problems of machine learning [11].
On the other hand, when fluctuations tend to zero, i.e. in the semiclassical limit, the dynamics of the DQM goes into a more complex initial smooth dynamics. The old problem – the equivalence of structural stability and hyperbolicity for smooth discrete dynamical systems [12] is established by this way in this paper.
The paper goal is 1) to build the foundations of the theory of dynamic quantum models (DQM); 2) to demonstrate the application of this theory for computer research of concrete systems and for solving traditional problems of the theory of smooth dynamical systems.
The paper is organized as follows: in part 2 we synthesize the dynamic quantum models (DQM), in section 2.2 we define the DQM attractor, show the uniqueness of this definition and establish properties of the DQM attractor; in part 3 we show that structurally stable realizations of DQM are dense and open on the set of all its realizations; in part 4 we demonstrate the use of computer modeling for rigorous proof of hyperbolicity of the attractor of Henon system; part 5 concludes.
We had to omit proofs of some theorems in order to fit the paper format. 2
The Dynamic Quantum Model: Basic Definitions
Let p(x) be an n -dimensional smooth vector field on an n -dimensional smooth
Riemannian manifold M , where x(x1, x2 ,..., xn ) are local Euclidean coordinates on
M , pi (x) C (Rn ) ( i 1,..., n ). On each phase curve
x(t) M
of the
dynamical system generated by this vector field
dxi pi (х) , ( i 1,..., n )
dt
(
So, the distance d traveled by a point along the path of (
2 ‖ ‖2 , = ‖ ‖∆ = 2 ‖ ‖
Here by assumption t is the minimal time interval after which it becomes possible
to make a new measurement, the difference from which will exceed the error, i.e. get a
significantly different measurement. Owing to (
2 t p( ) d s(t) . Thus 1) the time interval between the 0 nearest significant measurements is unchanged on the optical time scale and is equal to 2 . (In other words, the distance between them in the metric is equal to 2 ). 2) During this time “white noise” (t) generates an irremovable random error, the standard deviation of which is equal to the distance d between the nearest significant measurements along the trajectory.
So, a dynamic quantum model first shifts each point along the phase curve of a
given dynamic system over the optical time 2 (or ρ – length 2 ), and then randomly
shifts on a distance not exceeding the length of the trajectory from the original to the
new point. The following rigorous definition summarizes this description. The
definition of a dynamic quantum model is given for an arbitrary dynamic system (
Let G be the shift map along the phase trajectories of (
M q( y, z)dz 1,
M zq( y, z)dz Gy d ( y) , where d ( y) 0 is a continuous function on M . Here q( y, z) defines the density of “local random dissipation caused by white noise,” the numbers d ( y) are assumed to be small. Of course, the function q( y, z) can also be assumed continuous, approximating it on M with a smooth function for any given accuracy. Then Definition 1. The Markov process with the transition function
A
P( y, A) q( y, z)dz ( A M )
is called the dynamic quantum model (DQM) for the dynamic system (
Thus, based on the differential equations (
Definition 2. Let i be cells with a diameter of some partition of the phase space of a dynamical system and 0 is the initial state. Then the Markov chain with transition probabilities from i to j equal to pij = 1
P( y, j ) d 0 will be called 0 (i ) yi the - discretization of DQM with transition function P( y, A) and initial state 0 .
Attractor is the key concept of the theory of dynamical systems; its physical meaning is that it is “the space of steady-state regimes”. The point of the phase space is contained in the attractor if it belongs to the carrier of the “stationary state of the system”, i.e. to a measure not changing over time.
Let M be a compact phase space, P is some DQM on M .
Definition 3. The probability measure on M will be called the stationary (equilibrium) state of DQM if P( ) . The DQM attractor is the union of the carriers of all stationary states.
Theorem 1. (Perron-Frobenius theorem for DQM). Let M be an invariant closed set of DQM P that does not contain its own invariant closed subsets (that is minimal with respect to P ). Then 1. there is a unique stationary state , whose carrier is . The state is ergodic (that is the flow P is ergodic with respect to measure ). 2. For any other state (probability measure) on lim 1 n Pk = . n n k 1 3. If is a probabilistic stationary measure of some - discretization of the given DQM on then lim = .
0
Proof. Let M be an invariant closed set of DQM that does not contain its own
invariant closed subsets. Let be a stationary measure of some discretization of the
given DQM on with cells of diameter (that is, a probability invariant measure of
a Markov chain defined by Definition 3). On a compact subset of the phase space
the set of probability measures R R() forms a convex metrizable compact in the
weak topology. Therefore in any sequence of measures k one can find a
subsequence n , converging to some measure from R : nlim n = R in the
sense of the weak topology on R . Since P n n n00 (in the sense of the
weak topology) by virtue of definition 3, then P i.e. is a stationary state of
DQM. Since by the condition does not contain non-empty proper invariant subsets
of DQM (i.e. it is metrically transitive), then for any P -invariant measure on the
ergodic Neumann theorem holds: for any continuous function f on
1 n
L2 lim f (Pk ) fd
n n k 1
(
Obviously, there are only finitely many components of the DQM attractor k on M , that is such invariant subsets of the attractor that do not contain proper invariant non-empty subsets. On each component k of the DQM attractor there is a unique probability invariant measure k : P k k . The density of k is positive on the interior of k by the definition of DQM. Any stationary state on M is a convex combination of stationary states k on k .
Let G be shift map along the phase trajectories of (
Definition 4. For the DQM trajectory for the time tn : y0 , y1,..., yn its differential is
Dn ( ) DG( yn ) ... DG( y1) DG( y0 ) where DG( yk ) is the differential of G at the point yk , k n 0,1,... . For the DQM trajectory : yn ,..., y1, y0 differential Dn ( ) DG( yn ) ... DG( y1) DG( y0 ) at n 0,1,... .
The measure , induced by the measure in accordance with the Kolmogorov theorem [13], is defined on the space of DQM trajectories on the component of the DQM attractor.
Theorem 2. Let M is a component of the DQM attractor of dimension
m dim M . Then for DQM with sufficiently small d min d ( y) (where
y
d ( y) 0 are constants from (
u Rm ( u 1) there are limits
DQM trajectories
at any nonzero vector nlim 1n ln Dn ( )u r , where r 1,2,..., s m dim M . 2. At each point of each such trajectory , the filtering of subspaces is uniquely defined: forward L1 ( y) L2 ( y) ... Ls ( y) Rm and back Ls ( y) ... L2 ( y) L1 ( y) Rm , associated with the numbers 1 2 ... s so that lnim 1n ln Dn ( )u r u Lr ( y) and u Lr1 ( y) , lim 1 n n
ln Dn ( )u r u Lr ( y) and u Lr1 ( y) . and yn1 are consecutive points of the trajectory
These filtrations are invariant with respect to the DQM differential. Exactly if yn at times tn and tn1 respectively then the differential DG( yn ) translates the filtering at the point yn in the filtering at the point yn1 .
Proof. Consider DQM on as a random process X (t, ) , where t is discrete time, t tk , k 0,1,2,..., is a DQM trajectory. Namely, for k M let η = (..., k ,..., 1, 0 ,1,..., k ,...) . Then the DQM trajectory (t, y0 ) with an initial point y0 M is the sequence X (t0 , ) y0 , X (t1, ) y1 Gy0 0 , X (t2 , ) y2 Gy1 1 , …, X (tk , ) yk Gyk 1 k 1 ,… . (Here d is assumed to be so small that the addition of Gyk 1 k 1 when k d performed on the local map of the manifold M in R m ). Thus, the DQM trajectory is defined uniquely by a sequence of vectors η and an initial point y0 : ( y0 , η0).
On the set of DQM trajectories X (t, ) on induces the dynamic process T – the trajectory of the trajectories: T0 1 , T1 2 , …, T k 1 k , … . Namely if 0 ( y0 , η0), where y0 X (t0 ,0 ) , η0 (..., k ,..., 1,0 ,1,..., k ,...) and 1 ( y1, η1), then y1 Gy0 0 X (t1,0 ) , η1 = R η0, where R is shift operator to the right. If 2 ( y2 , η2) then y2 Gy1 1 X (t2 ,0 ) , η2 = R η1; in the general case for k ( yk , ηk) we get yk Gyk1 k1 X (tk ,0 ) , ηk = = R ηk-1. By the Kolmogorov theorem on the set of DQM trajectories on the probability measure was determined, induced there by a stationary state on . By construction measure inherit from measure the invariance with respect to T ( (T ) ) and ergodicity of T (i.e. its metric transitivity) under measure .
Let a(n, ) Dn ( ) for ( y0 , η). Then a(n, ) are measurable functions on a probability space with measure and a(n k, ) a(k,T k ) . This means that the square matrices a(n, ) of order m are a multiplicative cocycle on the space of trajectories with respect to its automorphism T by the definition of the cocycle [11]. Since G is a diffeomorphism, then DG( y) 0 for all y , whence ln( DG( y) ) is continuous function on compact and ln DG( y) d . On the other hand by definition a measure for any open y subset C with the characteristic function C C d ({ ( y, ) y C}) = ({y y C}) = C d .
M Therefore for any piecewise continuous function g on M g d = g d . In M particular since a(0, ) DG( y) on each trajectory ( y, η), then ln a(0, ) d = ln DG( y) d . This inequality means that the cocycle y a(n, ) is Lyapunov and this is the condition under which the multiplicative ergodic theorem for this cocycle holds.
This theorem asserts that almost all trajectories under measure are Lyapunov correct. This means, in particular, that 1. for such with u Rm ( u 1) there are limits lim 1 ln a(n, )u r ( ) , n n where r 1,2,..., s s( ) m . 2. On each such trajectory , the filtering of subspaces is uniquely defined: forward L1 ( y) L2 ( y) ... Ls ( y) Rm and back Ls ( y) ... L2 ( y) L1 ( y) Rm , associated with the numbers 1 2 ... s ( s s( ) ) so that lnim 1n ln a(n, )u r u Lr ( y) and u Lr1( y) , nlim 1n ln a(n, )u r u Lr ( y) and u Lr1( y) .
These filtrations are invariant with respect to the automorphism T : if Tn n1 , then the cocycle a(n, ) takes the filtration n to the filtration n1 .
Since by the Kolmogorov theorem flow T on a probability space with a measure inherits ergodicity from ergodicity P on with a measure , which was established in Theorem 1. Then the values of r ( ) r , s( ) s coincide for almost all DQM trajectories under measure . In view of the correspondence a(n, ) Dn ( ) the theorem immediately follows from here, QED.
By analogy with the theory of smooth dynamical systems the numbers r we will call the Lyapunov characteristic exponents of the component of the DQM attractor. 3
Structural Stability in DQM
Definition 5. The DQM realization is a sequence of smooth mappings Gk ( y) on
in M ( k 0,1,2,... ) if Gk ( y) G( y) C1 d ( y) d , where d(y)
are the constants from (
Here all the maps Gk ( y) are diffeomorphisms on Λ in Λ for sufficiently small d . In terms of content (tk , y) = Gk ( y) G( y) are small random deviations caused by “white noise” at the point y at time tk . By definition, any DQM trajectory ( y0 , η): yk Gyk1 k ( k 0,1,2,... ) is given by the initial point y0 M and sequence of deviations η (..., k ,..., 1, 0 ,1,..., k ,...) . But on the DQM realization the function of deviations (tk , y) is fixed; therefore, on it the DQM trajectory with the initial point y0 is uniquely determined: ( y0 ) .
Definition 6. A DQM realization Gk ( y) ( k 0,1,2,... ) on a compact set K will be called a hyperbolic realization of DQM if at each point y Kk , where K0 K,Gk (Kk1) Kk there exists a decomposition of the tangent bundle TKk into the Whitney sum of the subbundles Eks ( y) and Eku ( y) : TKk = Eks ( y) + Eku ( y) , satisfying the following conditions:
DGk (Eks ) Eks1,
DGk (Eku ) Eku1 ; 2. DGk compresses Eks : on every trajectory with an initial point y Kk at the time moment tk there are such constants b 0 and (0 1) that for any u Eks and any natural n
Dn ( )u bn u ( u Eks ( y) ). 3. DGk stretches Eku ( y) , more precisely, on each trajectory with an initial point y Kk at the time moment tk for any u Eku and a natural n
Dn ( )u 1 u ( u Eku ( y) ).
bn
Theorem 3. Hyperbolic realizations are everywhere dense on the set of DQM realizations. More precisely, for any DQM realization Gk ( y) ( k 0,1,2,... ) and for any sufficiently small 0 there exists such hyperbolic realization Gk ( y) of this DQM on the compact K , that 1. ( / K K / ) for the probabilistic invariant DQM measure on ; 2. on K k Gk ( y) Gk ( y) C1 ( k 0,1,2,... ).
Definition 7. The realization of the DQM Gk (x) on a compact K and the ~ realization Gk (x) of this DQM on a compact K~ are topologically equivalent if they are conjugate by means of homeomorphisms H k defined on some neighborhoods of the compacts Kk , where Gk (Kk 1 ) Kk , G~k (K~k ) = K~k 1 , K0 K , ~ ~ ~ K0 K : Gk H k H k 1 Gk ( k 0,1,2,... ).
Hk ,
Kk+1
Definition 8. A DQM realization Gk ( x) on a compact K is structurally stable if any realization of this DQM sufficiently close to Gk ( x) in C1 topology for all k 0,1,2,... is topologically equivalent to it.
In more detail: If for every point x Kk there are numbers dk (x) 0 such that
G~k ( x) from Gk (x) G~k (x) for any realization of this DQM C1 ≤ dk ( x) for all k ~ and x Kk the topological equivalence of the realizations Gk and Gk follows, then Gk is structurally stable.
If all Gk ( x) and all compacts K k coincide for all k , then we obtain the definition of the structural stability of a diffeomorphism.
Theorem 4. Any hyperbolic realization of the DQM Gk on the compact set K is structurally stable.
Corollary 1. A diffeomorphism G on a compact manifold M is structurally stable in the sense of Definition 8 if and only if it is (non-uniformly) hyperbolic.
Proof. Let a diffeomorphism G be structurally stable. By Definition 8, this means
that for some d (x) d from G(x) G~(x) C1 d (x) ( x M ) it follows
~
that the diffeomorphisms G and G are topologically equivalent, that is, are conjugate
on M by means of a homeomorphism. Consider DQM for G with the same d (x) in
(
Conversely, the fact that the non-uniform hyperbolicity of a diffeomorphism G onto M implies its structural stability in the sense of Definition 8 directly follows from Theorem 4, QED. arbitrarily small perturbations. However from
~ follows that G and G are topologically equivalent by the condition of the corollary. Therefore G is uniformly hyperbolic on M , QED.
~ G(x) G(x) C1 ( x M ) follows
Example of DQM Application: Henon System Attractor For the two-dimensional M. Henon system [10]: (x, y) (1 y ax2 ,bx) values of parameters a 1.7, b 0.5 are chosen such that the hyperbolicity of dynamics on the attractor with this parameters is rigorously proved for R. Lozi system [14] (x, y) (1 y a x ,bx) . The proof of the hyperbolic dynamics here is based on the following statement, specifically focused on the study of concrete dynamical systems. For ease of application to the Henon system in the formulation we restrict ourselves to the two-dimensional case, although the multidimensional generalization is also true.
Corollary 3. Let i be the cells of -discretization of the DQM attractor for the system given by the diffeomorphism G , xi i (1 i N ). Let the eigenvalues 1(xi ) and 2 (xi ) of the differential DG at each point xi i (1 i N ) satisfy the conditions 1(xi ) , 2 (xi ) for some ( 0 1) and 1
(1 )2
4(4 G
2
1)
,
(
is the norm of G in C 2 . Then 1. the initial system given by the diffeomorphism G is hyperbolic on its attractor; 2. any DQM - realization of this system is hyperbolic on the DQM attractor and is topologically equivalent to the initial system; 3. the support of the attractor of the initial hyperbolic system and the attractor of its
DQM - realization coincide with an accuracy of order .
The proof of this statement essentially reproduces the proof of Theorem 4, estimate (
On the basis of outcomes of the numerical researches, visually presented in Figure 2, we choose a rectangle {(x, y) | 1 x 1.5; 0.1 y 0.1} . In next Figure 3 for each iteration t 1,2,...,500 of Henon system corresponds its coordinate x(t) on the ordinate axis.
The animation in Figure 3 suggests that the system is hyperbolic. 2. The Prestep procedure splits the rectangle into cells i - squares with sides of length 0.01 parallel to the axes of coordinates ( 1 i N ).Then each cell i Prestep associates a set of cells into which points from i can fall into one step of the dynamics of the Henon system. In this case it is formally verified that the domain is indeed invariant with respect to the discretization of the DQM, given by the constructed partition of . In other words Prestep defines a topological Markov chain H , the state space of which is the set of cells i . 3. The Findattr procedure finds in the attractor of a topological Markov chain H defined in Prestep. Its algorithm is based on the following consideration. On the state space {i } consider a transitive quasi-order relation: i j if there exists a trajectory H from i to j . The state i is recurrent if i i . Recurrent states are divided into equivalence classes: i ~ j i j i . On {i } H () H 2 () H 3 () ... H n () . If H n () H n1 () then H n () is the DQM attractor. In the case under consideration, the attractor turns out to be connected, which corresponds to Fig. 2, obtained by the Animate procedure. 4. The Hyperproc procedure performs a main check: do the conditions of Corollary 3 be satisfied on the attractor found by Findattr? For the Henon system under consideration on a rectangle {(x, y) | 1 x 1.5; 0.1 y 0.1} we obtain
G
2
max{ (2ax)2 b2 1, 2a} 6.1. The Hyperproc procedure
establishes that for the differential
DG eigenvalues
1(xi ) 0.4
and
2 (xi ) 1.7 for all xi . The value 1/1,7 0.59 . Thus 0.59 ;
i
however, we choose the value 0.7 with a margin. Then, in accordance with (
Now the cell length of i (the length of a square with sides parallel to the axes of
coordinates) is chosen equal to 0.0005 (1 i N ) and already for such a small
partition of the rectangle we repeat the Prestep Findattr Hyperproc cycle
described above. Now the other smaller cells are i , the other xi i and the other
eigenvalues 1(xi ) and 2 (xi ) respectively (1 i N ). If now again 1(xi ) 0.4
and 2 (xi ) 1.7 holds for all i , then (
The selected values of the parameters a 1.7, b 0.5 are not the only ones. For example, similar results are obtained for a 1.4, b 0.35 . 5
Conclusion The structural stability of a mathematical model is a necessary condition for its correctness. It is also necessary for applicability of numerical methods, computational experiments since they inevitably lead to errors.
But after S. Smale's works it became clear that in smooth dynamics the system of a general form is not structurally stable and therefore there is no strict mathematical basis for modeling and computational analysis of systems. The contradiction appeared in science: according to physicists dynamics is simple and universal.
The paper proposes a solution to this problem based on the construction of dynamic quantum models (DQM). DQM is a perturbation of a smooth dynamical system by a Markov cascade (time is discrete). The dynamics obtained in this way are simpler than the classical smooth dynamics: the structurally stable realizations of DQM are everywhere dense (Theorem 3) and open (Theorem 4) on the set of all DQM realizations. This dynamics has a clear structural theory: unlike the classical systems, the DQM attractor is uniquely defined (Theorem 1), Lyapunov exponents exist for any DQM (Theorem 2).
As a Markov cascade, the DQM is approximated by a Markov chain and on a compact set by a finite Markov chain arbitrarily exactly (Theorem 1). This allows you to clearly understand the DQM dynamics and build effective algorithms for the study of concrete systems that are always oriented towards parallel computing and do not require stable (according to Hadamard) solutions. For example, in part 4 we demonstrate the use of computer simulation for rigorous proof of hyperbolicity of the attractor of Henon system.
On the other hand, when fluctuations tend to zero, i.e. in the semiclassical limit, the dynamics of the DQM goes into the initial smooth dynamics. In part 3 the equivalence of structural stability and hyperbolicity for smooth discrete dynamical systems is established along this path (Corollaries 1 and 2).
In the future, we intend to apply the DQM algorithms, that oriented towards parallel computing and do not require stable solutions, to traditional problems of computational methods.
We also intend to generalize dynamic quantum models on dynamical systems that using logical operations: proofs of theorems, software applications, information and network systems, etc. A natural and even obvious implementation tools for such a generalization are the specialized neural network. This will allow the use of DQM methods for problems of artificial intelligence: identification, prediction, filtering, etc.