USING CARLEMAN EMBEDDING TO DISCOVER A SYSTEM S MOTION CONSTANTS

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1 (Preprint) AAS USING CARLEMAN EMBEDDING TO DISCOVER A SYSTEM S MOTION CONSTANTS John E. Hurtado and Andrew J. Sinclair INTRODUCTION Although the solutions with respect to time are commonly sought when analyzing the behavior of dynamic systems, there are other expressions that can be as revealing and important. Specifically, motion constants, which are time independent algebraic or transcendental equations involving the system states, can provide a broad view of a system s motion. Locally, at least, the motion constants of autonomous systems exist. Nevertheless, for finite dimensional, autonomous, nonlinear systems, the motion constants are generally difficult to identify. The motion constants for every finite dimensional, autonomous, linear system, regardless of its dimension, however, have recently been found. This paper attempts to discover the motion constants of nonlinear systems through the motion constants of a linear representation. The linear representation is achieved through a Carleman embedding, which provides a procedure to transform a finite dimensional system of analytic ordinary differential equations into an infinite system of linear differential equations The solutions of dynamic systems are commonly presented in terms of collections of integrals. These integrals can be either explicitly time dependent or time independent. Time-independent integrals are able to describe the shape of the state-space trajectories and provide a broad picture of the motion irrespective of time. A complete set of time-independent integrals reduces the solution process to finding the speed along these trajectories [Contopoulas 2002]. Here, time-independent integrals are referred to as motion constants. These are algebraic or transcendental equations involving the system states. Sinclair and Hurtado [2012] have recently found exhaustive solutions for the motion constants of every finite dimensional, autonomous, linear system. Significantly, a central feature of their process is the single motion constant for a 2 2 system. For an n-dimensional linear system, the n 1 motion constants are constructed through a coordinate transformation to a Jordan canonical form. The diagonal block structure of the Jordan form allows one to identify motion constants that are internal to each block by using patterns learned from simple 2 2 groups. The remaining motion constants of the entire system are found by relating the time behavior of separate blocks to one another. With the exception of the case of incommensurate oscillators, the procedure provides a complete set of motion constants. Carleman showed in 1932 that a finite dimensional system of analytic ordinary differential equations can be embedded within an infinite system of linear differential equations [Steeb 1989, Kowalski and Steeb 1991]. The Carleman embedding process is well understood, and researchers have Associate Professor, Aerospace Engineering Department, Texas A&M University, 3141 TAMU, College Station, TX Associate Professor, Aerospace Engineering Department, Auburn University, 211 Davis Hall, Auburn, AL

2 answered important questions like, what is the relationship between the solutions of a finite dimensional nonlinear system and the solutions of its infinite dimensional linear representation? Here, the Carleman embedding process is investigated further yet, by seeking to understand the relationship between the motion constants of a finite dimensional nonlinear system and the motion constants of its infinite dimensional linear representation. INTEGRALS AND MOTION CONSTANTS OF DYNAMIC SYSTEMS Nonlinear Autonomous Systems An nth-order autonomous dynamic system describes the evolution of a state vector x(t) R n. ẋ = f(x) (1) This system of differential equations may be solved by finding n functions of the states and possibly time that are constant along the trajectory associated with an initial condition x 0. Such functions are called integrals [Szebehely 1967]. φ k (x(t), (t)) = φ k (x 0, 0) for t R, k = 1,..., n (2) Although the value of each φ k is constant along the motion, evaluating these integrals may require knowledge of the current states and time. Separately, an nth-order autonomous dynamic system can have up to n 1 time-independent integrals. ψ k (x(t)) = ψ k (x 0 ) for t R, k = 1,..., n 1 (3) Because each ψ k is constant, its Lie derivative along the direction of motion is zero. ψ k = ( ψ k / x) T ẋ = ( ψ k / x) T f(x) = 0 (4) This equation represents a partial differential equation for an unknown time independent function ψ k ; the ψ k are called motion constants. Although Frobenius theorem conveniently guarantees the existence of n 1 local solutions, determining the motion constants for nonlinear autonomous systems can be challenging. Linear Autonomous Systems Linear autonomous systems are a special class of eq. (1). ẋ = Ax (5) These systems are completely integrable in the sense of possessing n independent analytic integrals. φ(x, t) = exp At x (6) But these systems also possess motion constants ψ k (x), and their explicit forms have recently been discovered [Sinclair and Hurtado 2012]. A discussion of these forms can not be adequately addressed in this note; nevertheless, the important aspects are that the motion constants of n- dimensional linear systems can be assembled in a systematic way using Jordan canonical forms and a thorough comprehension of the single motion constant of 2 2 systems. 2

3 CARLEMAN EMBEDDING In simple terms, Carleman embedding is a process wherein a finite dimensional nonlinear system is mapped to a higher (perhaps infinitely higher) dimensional linear representation via coordinate transformations. A popular illustrative example is the rate equation [Kowalski and Steeb 1991]. Suppose new coordinates v are introduced. ẋ = x + x 2 (7) v 1 = x, v 2 = x 2,..., v n = x n (8) The differential equations that govern the new coordinates can be computed. v n = n x n 1 ẋ = n x n 1 ( x + x 2 ) = n x n + n x n+1 = n v n + n v n+1, n = 1, (9) The single nonlinear autonomous system has been mapped to an infinite collection of linear autonomous systems. The behavior and characteristics of the new infinite dimensional linear system must reflect the behavior and characteristics of the original finite dimensional nonlinear system in some way. Here, we pursue an understanding of the nonlinear system through the motion constants of the new linear representation; this pursuit is hopeful because the motion constants of every autonomous linear systems are known. At its heart, the Carleman embedding technique represents an overparameterization of the original system. We suspect that motion constants of the original system are captured in the overparameterization; more specifically, we suspect that the motion constants are captured at some finite truncation. This thinking leads one to ponder over the meaning of any extraneous motion constants. This brings to mind that one quality of overparameterizations is that natural constraint relationships are created between the new states [Hurtado and Sinclair 2011], thus we suspect that the extraneous motion constants reflect these natural constraints. A ROTATIONAL KINEMATICS INVESTIGATION The idea of discovering the motion constants of a nonlinear autonomous system through a coordinate embedding is examined through an example. Consider the rotational kinematics of a rigid body wherein the body-fixed components of the angular velocity vector are known constants. This system takes the form of a nonlinear autonomous system if the classic Rodrigues parameters are selected as the attitude coordinates. ẋ = f(x) ẋ = x2 1 x 1 x 2 x 3 x 1 x 3 + x 2 x 2 x 1 + x x 2 2 x 2 x 3 x 1 ω (10) 2 x 3 x 1 x 2 x 3 x 2 + x x 2 3 Consider an overparameterization that maps the three current coordinates x to four new coordinates β. [β 0, β 1, β 2, β 3 ] = [1, x 1, x 2, x 3 ]/ 1 + x T x (11) The kinematics in terms of the new coordinates are linear. β = Aβ where A = ω 1 ω 2 ω 3 ω 1 0 ω 3 ω 2 ω 2 ω 3 0 ω 1 ω 3 ω 2 ω 1 0 (12) 3

4 This state matrix has four eigenvalues, λ 1,2 = ±ı ω T ω/2, λ 3,4 = ±ı ω T ω/2, with four independent eigenvectors. The substitution ω 0 ω T ω is helpful in the ensuing equations. Following the process laid out by Sinclair and Hurtado, a Jordan canonical form is formed via A = MJM 1 where M is composed from the eigenvectors and J contains 2 2 blocks of zeros and the imaginary parts of the eigenvalues on the sub and super diagonals. M = ω 0 ω 2 /(ω1 2 + ω2 2 ) ω 1ω 3 /(ω1 2 + ω2 2 ) ω 0ω 1 /(ω1 2 + ω2 2 ) ω 2ω 3 /(ω1 2 + ω2 2 ) ω 0 ω 1 /(ω1 2 + ω2 2 ) ω 2ω 3 /(ω1 2 + ω2 2 ) ω 0ω 2 /(ω1 2 + ω2 2 ) ω 1ω 3 /(ω1 2 + ω2 2 ) J = ω ω ω ω 0 0 (13) (14) The matrix M is used to define Jordan coordinates y = M 1 β which are governed by ẏ = Jy. The Jordan system is partitioned so that motion constants are easily constructed. For this example, the three independent motion constants in terms of the four Jordan coordinates are given as follows. ψ 1 = y y 2 2 ; ψ 2 = y y 2 4 ; ψ 3 = y 2y 3 y 1 y 4 y 1 y 3 + y 2 y 4 (15) Another motion constant that is not independent of these three is ψ 0 = y T M T My. This motion constant is evident from eq. (11) which produces 1 = β T β. Consequently, any three of the four ψ k motion constants form an independent set. Judiciously selecting ψ 0, ψ 1, and ψ 2 and writing these in terms of the overparameterized β coordinates gives the following. ψ 0 = β β β β 2 3 (16) ψ 1 = ω 2 0β (ω 2 β 0 ω 3 β 1 + ω 1 β 3 ) 2 (17) ψ 2 = ω 2 0β (ω 1 β 0 + ω 3 β 2 ω 2 β 3 ) 2 (18) These can also be written in terms of the original (minimal) x coordinates. ψ 0 = 1 (19) ψ 1 = ω2 0 x2 2 + ω2 1 x2 3 + ω2 2 + ω2 3 x2 1 2ω 1ω 3 x 1 x 3 2ω 2 ω 3 x 1 + 2ω 1 ω 2 x 3 ω 2 0 (1 + x2 1 + x2 2 + x2 3 ) (20) ψ 2 = ω2 0 x2 1 + ω2 1 + ω2 2 x2 3 + ω2 3 x2 2 2ω 2ω 3 x 2 x 3 2ω 1 ω 2 x 3 + 2ω 1 ω 3 x 2 ω 2 0 (1 + x2 1 + x2 2 + x2 3 ) (21) One can confirm that functions ψ 1 and ψ 2 are constant along the trajectories of the dynamic system given by eq. (10), ( ψ k / x) T f(x) = 0. These equations illustrate that three independent motion constants for the overparameterized fourcoordinate system reflect two independent motion constants for the minimal three-coordinate system plus an extraneous motion constant that reflects the natural constraint involving the excess parameters. Two things to note, however, are that the coordinate embedding in this case is exact (i.e., a linear representation is exactly achieved with a finite overparameterization), and the natural constraint (1 = β T β) was not precisely found to be a motion constant of the linear representation, although it is functionally dependent on the extracted motion constants. 4

5 A ROTATIONAL DYNAMICS The examination of discovering the motion constants of a nonlinear autonomous system through a coordinate embedding is continued through a second example. A classic nonlinear autonomous system is the torque-free rotational dynamics of a rigid body. ω = f(ω) ω 1 = a 1 ω 2 ω 3 ; ω 2 = a 2 ω 1 ω 3 ; ω 3 = a 3 ω 1 ω 2 (22) Here, ω are the body-fixed components of the angular velocity vector and a k represent the rigid body inertia parameters. a 1 = (I 2 I 3 )/I 1 ; a 2 = (I 3 I 1 )/I 2 ; a 3 = (I 1 I 2 )/I 3 (23) It is well-known that this system has two time-independent integrals, which are the rotational kinetic energy and the magnitude of the angular momentum. T = 1 2 I 1ω I 2ω I 3ω 2 3 ; h 2 = I 2 1ω I 2 2ω I 2 3ω 2 3 (24) The Carleman embedding process to convert the three-dimensional nonlinear system to an infinitedimensional, overparameterized, linear system begins with the definition of new coordinates. x (ijk) = ω i 1 ω j 2 ωk 3, i, j, k = 1,..., (25) According to this definition, there are three x coordinates that are linear in the ω states; and six x coordinates that are quadratic in the ω states; and ten x coordinates that are cubic in the ω states; or (p + 1)(p + 2)/2 coordinates that are order p in the ω states. The new coordinates are governed by a system of linear autonomous equations. ẋ (ijk) = i a 1 x (i 1 j+1 k+1) + j a 2 x (i+1 j 1 k+1) + k a 3 x (i+1 j+1 k 1) (26) Our approach to discover two motion constants for the original nonlinear system is outlined in the following steps: 1. Truncate the length of the new coordinates at some point, say m. The collection of truncated states is denoted as x. 2. Retain the upper m m block of A. This is the truncated state matrix A. 3. Create a Jordan canonical form via A = MJM 1 where M is composed from the eigenvectors and J contains eigenvalues on the diagonal and sometimes ones on the subdiagonal. 4. Use M to define Jordan coordinates y = M 1 x which are governed by ẏ = Jy. 5. Use the methods discussed in Sinclair and Hurtado (2012) to determine the motion constants for the Jordan coordinates, ψ = ψ(y). 6. Use y = M 1 x to write the motion constants in terms of the truncated linear states x. 7. Use x = x(ω) x to write the motion constants in terms of the original coordinates ω. At this point, specific inertia values are used to facilitate the computations; we select a tri-inertial case I 1 = 1, I 2 = 2, and I 3 = 3. Regardless of the truncation limit, an eigenanalysis of the matrix A reveals that all of its eigenvalues equal zero. Some of the zero eigenvalues generate an independent eigenvector whereas others generate an eigenvector that is linearly dependent on the 5

6 others. Sinclair and Hurtado (2012) have shown that a Jordan coordinate associated to a 1 1 Jordan block with a zero eigenvalue is itself a motion constant. Subsequently using y = M 1 x for each of these special Jordan coordinates gives an expression for the motion constant in terms of the coordinates x. In this example we ll look for true motion constants of the original nonlinear system within these special Jordan coordinates. For example, suppose we truncate the linear representation to the coordinates x that are no more than quadratic in the ω states. This creates a nine element vector x : x 100, x 010, x 001, x 110, x 101, x 011, x 200, x 020, x 002 The 9 9 matrix A has three 1 1 Jordan blocks with a zero eigenvalue. The Jordan form is arranged so that the special Jordan coordinates that are constant for all time are y 1, y 2, and y 3. These three Jordan coordinates are motion constants, and using M they can be mapped to motion constants in terms of x, which are assumed to honor the Carleman coordinate definitions in eq. (25); this then gives motion constants in terms of ω. ψ 1 = y 1 = x x 200 = ω 1 ω 3 + ω 2 1 (27) ψ 2 = y 2 = x x 020 = ω ω 2 2 (28) ψ 3 = y 3 = x x 001 = ω ω 2 3 (29) Computing the Lie derivative shows that these expressions in ω are not motion constants for the true nonlinear system, so this truncation level does not capture the true motion constants of the original system. Moreover, unlike the previous example, none of the eight motion constants at this truncation level reflect the coordinate constraints that are part of the overparameterized definitions (see eq. (25) which can be interpreted to read as x (ijk) = x i 100 xj 010 xk 001 ). Truncating the linear representation to the coordinates x that are no more than quintic in the ω states gives a total of fifty-five elements for x. The matrix A has fourteen 1 1 Jordan blocks with a zero eigenvalue. The Jordan form is arranged so that the special Jordan coordinates that are constant for all time are y 1 through y 14. These fourteen Jordan coordinates are motion constants, and they can be mapped to motion constants in terms of x or ω using M. Two of these fourteen motion constants read as follows: ψ 1 = x 020 3x x x 022 x x 220 (30) = ω 2 2 3ω ω ω 2 2ω 2 3 ω 2 1ω ω2 1ω 2 2 (31) ψ 2 = x x x x x 400 (32) = ω ω ω ω 2 1ω ω 4 1 (33) The Lie derivative of these expressions in ω confirm that these are motion constants for the true nonlinear system. Moreover, the rotational kinetic energy T and magnitude of angular momentum h 2, see eq. (24), are not independent functions from ψ 1 and ψ 2. Therefore, this truncation level captures the true motion constants of the original system. Although motion constants for the true nonlinear system have been identified, the extraneous motion constants do not seem to reflect the coordinate constraints that are part of the overparameterized definitions. This is because each level of truncation neglects the influence of higher order states in the governing differential equations. 6

7 ω 1 ω 2 ω 3 x 100 x 010 x 001 ψ 1 ψ Figure 1. Graphs of a representative simulation. The solid lines show ω from a nonlinear simulation; the dashed lines show x 100, x 010, x 001 for a truncated linear representation; and the dotted lines show the two motion constants when evaluated using the x ( ijk) states. Interestingly, finding the exact motion constants for the nonlinear system through the Carleman embedding does not mean the time evolution is exactly matched. An illustration is shown in Figure 1. The solid lines show a two second evolution of ω from simulating the nonlinear equations of motion. The initial conditions are ω = [1.1, 1.2, 1.3] T. The dashed lines show the evolution of x 100, x 010, x 001 for a 55 state linear representation. Neglecting the affect of the truncation, these three states represent ω in the linear representation. Notice that the dashed lines diverge from the solid lines before much time has elasped. Also shown as dotted lines are the two motion constants when evaluated using the evolution of the linear states x ( ijk). That is, the dotted lines are plots of eqs. (30) and (32). CONCLUSIONS An idea for discovering the motion constants of a nonlinear autonomous system is presented in this paper. The idea uses an overparameterization, or Carleman embedding method, to map a finite dimensional nonlinear system to a higher dimensional linear representation via coordinate transformations. The idea further uses the result that the motion constants for every finite dimensional, autonomous, linear system, regardless of its dimension, have recently been found. We have shown through examples that the idea has merit: in each case we were able to discover exact motion constants for the original nonlinear system using overparameterization. When the overparameterization or embedding is exact, then the extraneous motion constants reflect the coordinate constraints that arise through the overparameterized coordinate definitions. This is as it should be because the governing equations are exact differentials of the overparameterized definitions. When the overparameterization or embedding is inexact (because of truncation), then the extraneous motion constants do not necessarily reflect the coordinate constraints because the truncation produces an approximation of the true governing equations. Thus the governing equations are no longer exact differentials of 7

8 the overparameterized definitions. REFERENCES [1] G. Contopoulos, Order and chaos in dynamical astronomy. Springer, [2] J.E. Hurtado, A.J. Sinclair, Lagrangian mechanics of overparameterized systems, Nonlinear Dynamics, Vol. 66, [3] K. Kowalski and W.-H. Steeb, Nonlinear dynamical systems and Carleman linearization. World Scientific Publishing Co Inc, [4] A.J. Sinclair and J.E. Hurtado, The time-independent integrals of linear autonomous dynamical systems, in preparation. [5] W.-H. Steeb, A note on Carleman linearization, Physics Letters A, Vol. 4, No. 6, [6] V. Szebehely, Theory of Orbits, Sects. 1.3, 1.8 & 2.1. Academic Press,