The Second Lie-Group SOo ( n , 1 ) Used to Solve Ordinary Di ff erential Equations

Liu (2001) derived the first augmented Lie-group S Oo(n, 1) symmetry for the nonlinear ordinary differential equations (ODEs): ẋ = f(x, t), and developed the corresponding group-preserving scheme (GPS). However, the earlier formulation did not consider the rotational effect of nonlinear ODEs. In this paper, we derive the second augmented Lie-group S Oo(n, 1) symmetry by taking the rotational effect into account. The numerical algorithm exhibits two solutions of the Lie-group G ∈ S Oo(n, 1), depending on the sign of ‖f‖2‖x‖2 − 2(f · x)2, which means that the algorithm may be switched between two states, depending on x. We give numerical examples to assess the new algorithm GPS2, which upon comparing with the GPS can raise the accuracy about three orders. It is interesting that for the chaotic system the signum function sign(‖f‖2‖x‖2 − 2(f · x)2) is frequently switched between +1 and −1 in time.


Introduction
There are a lot of numerical methods developed to solve the ordinary differential equations (ODEs), which are effective for most situations.However, when the ODEs possess special structures, like the conserved integrals and the Lie-group symmetries, those numerical methods designed for the general purpose cannot maintain the special structures, unless we can design a particular algorithm to do that.There was a substantial development in the Lie-group geometric integrators for solving ODEs (Iserles, Munthe-Kaas, Nørsett, & Zanna, 2000;Hairer, Lubich, & Wanner, 2002).
The majority of the Lie-group integrator is for providing a good performance algorithm that can enforce the orbit of numerical solution on the Lie-group manifold, which is a key way to retain qualitatively correct behavior, and also to reduce numerical errors.At the present time, there are many famous geometric integration methods, like the Crouch-Grossman, the RKMK, the Magnus, and the Fer methods.The existent studies clearly indicated that the Lie-group methods can improve the qualitative behavior and for a long term integration.Basically, we have two approaches in the higher-order Lie-group integration methods.One is using the Lie group property and another is employing the Lie algebra structure, of which the geometric integrators developed by Hairer, Lubich and Wanner (2002), Iserles (1984), Iserles and Nørsett (1999), Iserles, Munthe-Kaas, Nørsett and Zanna (2000), Lee and Liu (2009), Munthe-Kaas (1998, 1999), and Zhang andDeng (2004, 2006) can be referred.
The remaining parts of this paper are organized and arranged as follows.In Section 2 we review the first augmented Lie-group S O o (n, 1) symmetry derived by Liu (2001), and introduce the group-preserving scheme (GPS).Section 3 explores the concepts from the Jordan algebra and the Jordan dynamics to help us for finding the Lie-group DS O(n) symmetry of nonlinear ODEs.The new developments of the second augmented Lie-group S O o (n, 1) symmetry are described in Section 4, while the explicit formulas for the Lie-group S O o (n, 1) integration method are derived in Section 5.In Section 6 we derive a novel explicit group-preserving scheme GPS2, which is used to solve nonlinear ODEs.In particular, we will emphasize the chaotic behavior of some well known chaotic systems as being observed from the new method.Finally, the conclusions are drawn in Section 7.

The First Augmented Lie-Group Symmetry
Let us consider ẋ which is an n-dimensional ODEs system for depicting an initial value problem.Generally speaking, the flow mapping given by φ t (x 0 ) = x(t; x 0 ) (2) satisfies φ t 1 • φ t 2 = φ t 1 +t 2 as being a one-parameter semi-group.Although the ODEs possess such a property of being a semi-group, but in general, the ODEs do not necessarily have the Lie-group structure.
For Equation ( 1) we can consider an orientation of state vector (Liu, 2001): where x := √ x • x is the length of x, and the dot in x • x signifies the inner product.We suppose that x 0.
Upon using Equations ( 1) and ( 3), x is depicted by Then, from Equations ( 1), ( 3) and ( 4) it follows that Therefore, we have decomposed Equation ( 1) for x into Equations ( 4) and ( 5), respectively, for x and n with x n = x.
From Equations ( 4) and ( 5) it follows that d dt which shows that the length x is an integrating factor of the governing Equation (5) for n.Then, inserting x n = x into the above equation and combining with Equation (4) we can write a Lie-form system: where X has one dimension higher than x, and is called the augmented state vector, satisfying where is the metric of the Minkowski space M n+1 .I n is an n × n identity matrix.
By employing the ODEs in Equation ( 7) with A ∈ so(n, 1), i.e., Liu (2001,2006) has developed a group-preserving scheme (GPS): where and h = Δt is a time increment.The values of x and f at a discrete time t = t k are simply denoted by x k and Because the above Lie-group S O o (n, 1) symmetry of Equation ( 7) is realized in the Minkowski space for the augmented state vectors, this kind symmetry may be termed the augmented Lie-group symmetry, and the Liegroup symmetry found by Liu (2001) is the first augmented Lie-group symmetry for Equation (1).
It is known that a matrix A of the real Lie algebra so(n, 1) of the Lorentz group S O o (n, 1) has the general form (Hong & Liu, 1999): where The matrix A in Equation ( 7) is a special case of Equation ( 16) with A s s = 0 and A s 0 = f/ x .In this paper, we attempt to derive the second augmented Lie-group symmetry for Equation (1), of which the Lie-algebra is in the form of Equation ( 16) with A s s 0. This new method can generate a new algorithm which is better than the GPS.The following ODEs system: is termed a Lie-form system if A satisfies the properties of Lie-algebra.

The Jordan Structure
From the Jordan structure (Iordanescu, 2007(Iordanescu, , 2009) ) we can give a new aspect of Equation ( 5), and then the second augmented Lie-group symmetry for Equation (1) can be deduced.To derive the novel Lie-symmetry, we will use the Jordan algebra and the triple system developed by Liu (2000), who was the first one to develop a dynamical system based on the triple-vector (y, z, u): The triplet y, z and u can be functions of x and t.Liu (2000) has shown that y • zu is a conservative term, while u • zy represents a dissipative term.
By using the Jordan dynamics in Equation ( 19) and using n • n = n 2 = 1, we can write Equation ( 5) to be Moreover, Equation ( 20) can be recast to where u ⊗ y is defined by (u ⊗ y)z = y • zu, which is a dyad of u and y.Due to the Lie-group symmetry for Equation ( 21) is S O(n).
The pair in Equation ( 21) is called a Jordan pair, and the following operation: is called the Jordan cross product of the Jordan pair, which is an extension of the usual cross product of vectors in three-dimensional space R 3 .
From Equations ( 4) and ( 21) and x = x n we have which can be re-formulated as where As shown by Liu (2013a), Equation ( 25) is equipped with a Lie-group DS O(n) symmetry, with Sx rendering a dilation/contraction and Wx causing a rotation.It is interesting that the coefficient matrices S and W are fully determined by the Jordan pair in Equation ( 22); they are, respectively, the inner product and the Jordan cross product of the Jordan pair.Recently, Liu (2013a) has developed a DS O(n) Lie-group algorithm based on Equation ( 25).

The Second Augmented Lie-Group Symmetry
By the definition of Equation ( 3), Equation ( 4) can be written as and then Equations ( 24) and ( 27) can be written together as Moreover, in terms of we can find three inherent structures about Equation (28): Cone: Lie-algebra: B ∈ so(n, 1), where g := diag(I n , −1) is a signature (n, 1) metric tensor, and G is the Lie-group generated from B, which can be expressed as In the above, G s s is an n × n matrix, G s 0 is an n × 1 matrix, G 0 s is an 1 × n matrix, and G 0 0 is a scalar.Corresponding to the first augmented Lie-group symmetry found by Liu (2001), the above Lie-group symmetry is the second augmented Lie-group symmetry in the Minkowski space M n+1 for Equation (1).
By using Equation (33) we can identify two types internal symmetries for n and x, respectively: where x 0 and n 0 denote, respectively, the initial values of x and n.It can be seen that the symmetry for n is more complex than that for x, because its symmetry is a projective type.On the other hand, the above two Lie-group transformations are different from the S O(n) for n as shown in Equation ( 21), and the DS O(n) for x as explored by Liu (2013a).While the Lie-group symmetry for n in Equation ( 34) is a projective type of S O o (n, 1), denoted by PS O o (n, 1), the Lie-group symmetry for x in Equation ( 35) is a Poincaré type: a Minkowskian rotation G s s x 0 follows by an Euclidean translation x 0 G s 0 .It can be seen that the augmented coefficient matrix B of Equation ( 28) is fully determined by the Jordan pair ( 22) with where the scalar in the symmetric part is the inner product of the Jordan pair, and the skew-symmetric part is the Jordan cross product of the Jordan pair.
Now we can observe that the above B is of the general form in Equation ( 16), as being a full Lie-algebra of so(n, 1).Upon comparing the above B with the coefficient matrix A in Equation ( 7): the matrix B includes a skew-symmetric part W, which is absent in A. It can take the rotational effect of nonlinear dynamical flow into account.
From Equation (24), by using the dyadic operation for the last term, we have Then the first term and the last term are cancelled out, which leads to This ODEs system is the most simple Lie-form representation of Equation (1).Liu (2013b) has developed a GL(n, R) Lie-group symmetry and used it to develop a Lie-group algorithm based on Equation (39).Up to here we have derived four types Lie-group symmetries about Equation (1).

Two Solutions of G
From Equation ( 28) we can develop a numerical scheme, by supposing that the Jordan pair is constant within a small time step h, where the bar means that they are taking values at a mid-point t ∈ [0, h].
From Equation (4), Liu (2013a) has written the ODE for the length to be Under the above assumption in Equation ( 40) we can obtain However, a more subtle representation of the ODE for the length is in terms of the Jordan pair (40): which is the second equation in Equation ( 28).Below, we will derive a new symmetry based on Equation (28).
For the special case with c 0 = 0 we have where ω = a , z 0 = a • x 0 and w 0 = b • x 0 .For this case x is a constant.
In Appendix A we give a detailed derivation of the solutions for (z, w, y) with c 0 0. Depending on the signum function of there exist two different types solutions of (z, w, y).We give the solutions of X(t) = (x(t), x(t) ) T and G in the following.

The Solution of X(t) When sign = +1
Next we consider the case with a 2 0 − 2c 2 0 > 0, of which by inserting Equation (A9) for z, w and y into Equation ( 47) and integrating it, render where in which ω = a 2 0 − 2c 2 0 , f 3 (t) = sin(ωt), and f 4 (t) = cos(ωt).By using the last equation in Equation (A9) and inserting z 0 = a • x 0 , and w 0 = b • x 0 we can obtain where To find G in Equation ( 33), which corresponding to a constant matrix B in Equation ( 36), is equivalent to find the augmented state transition matrix from X 0 to X(t).

The Solution of G When sign = +1
From Equations ( 56) and ( 58) we can obtain Besides the property in Equation ( 32), we can prove that which is important in the numerical scheme to preserve the orientation of x > 0. From Equation (A10) we have By using This ends the proof of Equation ( 61).

The Solution of G When sign = −1
For the case with a 2 0 − 2c 2 0 < 0 we have a similar G with The last inequality follows from by using Equation (A6).
Up to here we have derived two different G's as shown in Equations ( 60) and ( 62).For the ODEs: ẋ = f(x, t), depending on the value of sign( f 2 x 2 − 2(f • x) 2 ) we may alternatively use Equation ( 52) or Equation ( 56) to compute the solution of x(t) step-by-step.Hence, we have developed a two-phase Lie-group algorithm to solve Equation ( 1), where the algorithm may be switched between two states detected by the solution of x.

Numerical Examples
In order to distinct the present method from the group-preserving scheme (GPS) developed by Liu (2001), we may call the new algorithm to be the second GPS, simply denoted by GPS2, which is summarized as follows.
(i) Give an initial value of x 0 at an initial time t = t 0 and a time stepsize h.
(ii) For k = 0, 1, . .., we repeat the following computations to a specified terminal time t = t f : We may see that the most nonlinear ODEs fall into the first class with sign = +1 and use Equation ( 63) for the numerical integration, because in Equation ( 63) there are sin(ω k h) and cos(ω k h), while that in Equation ( 64) there are sinh(ω k h) and cosh(ω k h).However, for the chaotic system the situation is quite different, with Equations ( 63) and ( 64) both being used in the numerical integration.We use the following examples to assess the performance of the GPS2.

Example 1
Let us consider the following periodic system: Figure 1.For Example 1 with stepsize h = 0.1, the numerical errors of GPS2 and RK4 Figure 2.For Example 1 with stepsize h = 0.01, the numerical errors of GPS, DSON and GPS2 The exact solutions are x 1 (t) = 1.59941 sin t − 0.00004 sin 3t, x 2 (t) = 1.59941 cos t − 0.00012 cos 3t. (67) In Figure 1 we compare the numerical errors obtained by the GPS2 and the RK4, both using a stepsize h = 0.1.Under the stepsize h = 0.1, the GPS2 is better than RK4, while the GPS is failure.Under a small stepsize h = 0.01, these two schemes of GPS2 and RK4 have the same accuracy in the orders of 10 −9 − 10 −5 .In Figure 2 we compare the numerical errors obtained by the GPS2, the DS O(n) method (Liu, 2013a), and the GPS under the same stepsize h = 0.01, of which we can find that the DS O(n) and the GPS2 method is much better than the GPS with the accuracy being raised three to four orders.For this system, the signum function defined by Equation ( 51) is +1, because the solutions are bounded.

Example 2
Then we consider ẍ which can be recast to and has the solution:  showing the sign function

Duffing Oscillator
The Duffing equation under a forcing term is given by where α, β and γ are physical parameters, and f 0 and ω are, respectively, the amplitude and circular frequency of external force.As shown in Figure 5 Under a larger h = 0.05 the GPS2 still preserves the chaotic structure very well; but, the GPS is not succeeded, and is blowing after t = 800 sec.
In Figure 5(b) we show the signum function defined by Equation ( 51) for this chaotic system, which switches irregularly and fast between +1 and −1.

Chua's Circuit
In order to assess the performance of the GPS2, we compute the following numerical example of Chua's circuit (Chua, 1982(Chua, , 1986)): where h is a piecewise-linear function: However, the criterion of η depends on the time stepsize h used in the numerical integration.The time sequence of sign( f 2 x 2 − 2(f • x) 2 ) used to detect the chaos is independent to the time stepsize, which is fully determined by the vector field and the orbit of the given nonlinear dynamical system.For the chaotic systems investigated in this paper the time sequence of sign was frequently switched between +1 and −1.Since the new method is easy to be implemented numerically and has good computational efficiency and accuracy, it can be used to find the numerical solution of nonlinear ODEs.

)Figure 3 .
Figure 3.For Example 2 with stepsize h = 0.001, (a) the numerical errors of GPS and GPS2, and (b) showing the sign function We use h = 0.001.In Figure 3(a) the numerical errors obtained by GPS and GPS2 are compared.The accuracy of the above two methods are the same under a small time stepsize.In Figure 3(b) we show the signum function defined by Equation (51) for this system, which is regular and is changed from +1 to −1.For this system the solution x(t) = ln t is unbounded, such that after a certain time the signum function becomes −1.

Figure 4 .
Figure 4.For Example 3 with stepsize h = 0.002, the numerical errors of GPS and GPS2 (a) the numerical result is a chaos of the Duffing equation under the following parameters α = −1, β = 1, γ = 0.3 and Ω = 1.2, and f 0 = 0.32, where the time stepsize used is h = 0.05 and the time interval is t ∈ [0, 800].The transient part of the trajectory is starting from the initial point x 1 = 2 and x 2 = 0.

Figure 6 .
Figure 6.For Chua's circuit computed by the GPS2 under a stepsize h = 0.01, (a) the response, and (b) showing the sign function

Figure 9 .
Figure 9.For Rössler equations comparing the value of z and the sign function in time