Flow curvature method
| GINOUX Jean-Marc (2010), Scholarpedia, 5(11):10149. | doi:10.4249/scholarpedia.10149 | revision #142573 [link to/cite this article] |
Flow curvature method considers a trajectory curve, which is integral to any n-dimensional dynamical system, as a curve in Euclidean n-space. Then, the curvature of the trajectory curve, i.e. flow curvature may be analytically computed. The location of the points at which the flow curvature vanishes defines a manifold called flow curvature manifold. Since such a manifold only involves the time derivatives of the velocity vector field it has been established (Ginoux, 2009) that this knowledge is sufficient to describe the main characteristics of any n-dimensional dynamical system including:
- fixed points stability,
- invariant sets,
- centre manifold,
- normal forms,
- local bifurcations,
- slow invariant manifold, and
- integrability of dynamical systems.
One of the most important features of the Flow Curvature Method is the ability to compute the analytical equation of the slow invariant manifold of singularly-perturbed systems (i.e. systems comprising a small multiplicative parameter \(\varepsilon\) in their velocity vector field). Then, identity between the so-called Geometric Singular Perturbation Theory and Flow Curvature Method has been proven up to a suitable order in \(\varepsilon\ .\) Additionally, it has been stated that the flow curvature manifold associated with a two-dimensional dynamical system directly provides a first-order approximation in \(\varepsilon\) of the slow invariant manifold given by the Geometric Singular Perturbation Theory. Likewise, the flow curvature manifold associated with a three-dimensional dynamical system directly provides a second-order approximation in \(\varepsilon\ .\) Higher-order approximations may be simply obtained by replacing the flow curvature manifold by its successive Lie derivatives (time derivatives). Applications of the Flow Curvature Method to the Van der Pol and Lorenz models will shed more light on these aspects.
The main difference between the Flow Curvature Method and the Geometric Singular Perturbation Theory is that the analytical equation mentioned earlier is directly applicable to any n-dimensional slow-fast dynamical system, whether singularly perturbed or not (e.g. as shown in the Lorenz model). Moreover, the invariance of the flow curvature manifold follows the Darboux invariance theorem as opposed to the Fenichel's persistence theorem.
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n-dimensional dynamical systems
Let us consider a system of differential equations defined in a compact \(E\) included in \(\mathbb{R^n}\) with \(\vec{X} = \left[ (x_1 ,x_2 ,...,x_n) \right]^t \in E \subset \mathbb{R^n}\ :\)
\[\dot{\vec{X}} = \frac{d\vec{X}}{dt} = \vec{\Im} ( \vec{X} )\]
where \(\vec{\Im} (\vec{X}) = \left[ f_1 ( \vec{X} ),f_2 ( \vec{X} ),...,f_n ( \vec{X} ) \right]^t \in E \subset \mathbb{R^n}\) defines a velocity vector field in \(E\) whose components \(f_i\) are continuous functions \(C^\infty\) in \(E\) with values in \(\mathbb{R}\ ,\) bound by the assumptions of the Cauchy-Lipshitz theorem (see e.g. Coddinton et al. (1955), Guckenheimer et al. (1983) or Wiggins (1991)). A solution of this system is a parametrized trajectory curve \(\vec{X} (t)\) whose values define the states of a dynamical system. A dynamical system is said to be autonomous if each component of its velocity vector field does not explicitly depend on time. Since it can be shown that most non-autonomous dynamical systems can be transformed into autonomous systems, the Flow Curvature Method may also be applied to the majority of dynamical systems.
Flow curvature manifold
The location of the points, where the \((n-1)^{th}\) flow curvature integral of an n-dimensional dynamical system vanishes, defines an \((n-1)\)-dimensional flow curvature manifold described in the following equation:
\[\phi ( \vec{X} ) = \dot{\vec{X}} \cdot ( \ddot{\vec{X}} \wedge \stackrel{...}{\vec{X}} \wedge \ldots \wedge \vec{X}^{(n)} ) = \mbox{ det} ( \dot{\vec{X}}, \ddot{\vec{X}}, \stackrel{...}{\vec{X}}, \ldots , \vec{X}^{(n)} )=0\]
where \(\vec{X}^{(n)}\) represents the \(n^{th}\) time derivative of \(\vec{X}= \left[ x_1 ,x_2 ,...,x_n \right]^t\ .\)
Remark: The \((n-1)^{th}\) flow curvature is the highest degree possible. In the remainder of the text a flow curvature manifold will always refer to the highest degree flow curvature. In order to highlight one of the most important features of the Flow Curvature Method the article will specifically focus on a slow invariant manifold of slow-fast systems further on.
n-dimensional slow invariant manifold
The concept of invariant manifolds plays a very important role in the stability and structure of dynamical systems and especially for slow-fast dynamical systems and singularly perturbed systems. Since the beginning of the 20th century it has been subject to a wide range of seminal research. The classical geometric theory developed originally by Andronov et al. (1937), Tikhonov (1948, 1952) and Levinson (1950) stated that singularly perturbed systems possess invariant manifolds in which trajectories evolve slowly and toward which nearby orbits contract exponentially in time (either forward or backward) in the normal directions. These manifolds have been called asymptotically stable (or unstable) slow manifolds. Then, Fenichel (1971,1979) theory (independently developed by Hirsch et al. (1977)) for the persistence of normally hyperbolic invariant manifolds enabled to establish the local invariance of slow manifolds that possess both expanding and contracting directions, which were labeled as slow invariant manifolds as well. Thus, various methods have been developed in order to determine the slow invariant manifold analytical equation associated with singularly perturbed systems. The fundamental works of Wasow (1965), Cole (1968), O'Malley (1974, 1991) and Fenichel (1971, 1979) to name a few, gave rise to the Geometric Singular Perturbation Theory (see Jones (1994) and Kaper (1999)) and the problem of finding the slow invariant manifold analytical equation turned into a regular perturbation problem, in which one generally expected asymptotic validity of such expansion to breakdown according to O'Malley (p. 78 in O'Malley (1974) and p. 21 in O'Malley (1991)).
Later, it has been established (Ginoux et al., 2006), while using the framework of Differential Geometry that the slow invariant manifold associated with two- and three-dimensional dynamical systems was identically corresponding in dimension two to the curvature (first curvature) of the flow and in dimension three to the torsion (second curvature) of the trajectory curve. Then, this approach has been generalized to any n-dimensional slow-fast dynamical systems in Ginoux et al. (2008) and Ginoux (2009). As a result, the flow curvature manifold was defined with its slow manifold analytical equation, the invariance of which is established according to the Darboux invariance theorem.
Proposition 1 The flow curvature manifold of any n-dimensional slow-fast dynamical system directly provides its slow invariant manifold analytical equation.
Proof of this proposition can be found in Ginoux et al. (2008) and Ginoux (2009).
Darboux invariance theorem
Schlomiuk (1993) and Llibre (2004) have shown that in his memoir entitled Sur les équations différentielles algébriques du premier ordre et du premier degré, (Darboux, p. 71, 1878) had previously defined the concept of invariant manifold. Thus, Darboux invariance theorem may be stated as follows:
Proposition 2 The slow manifold defined by \(\phi ( \vec{X} ) =0\) where \(\phi\) is a \(C^1\) in an open set \(U\) is invariant with respect to the flow of the dynamical system if there exists a \(C^1\) function denoted \( k( \vec{X} ) \) and called cofactor which satisfies:
\[L_{\vec{V}} \phi ( \vec{X} )= k ( \vec{X} ) \phi ( \vec{X} ) \] for all \(\vec{X} \in U\)
and where \(L_\vec{V} \phi =\vec{V} \cdot \vec{\nabla} \phi = \sum_{i=1}^n \frac{\partial \phi}{\partial x_i} \dot{x_i} = \frac{d\phi}{dt}\ .\)
Proof.
In the location of the points where the functional Jacobian matrix J is locally stationary it is established that:
\[L_{\vec{V}} \phi ( \vec{X} ) = Tr ( J ) \phi ( \vec{X} ) = k ( \vec{X} ) \phi ( \vec{X} )\]
where \( k ( \vec{X} ) = Tr ( J ) \) represents the trace of the functional Jacobian matrix J.
For complete proof of this proposition see Ginoux et al. (2008) and Ginoux (2009).
Remark. Moreover, identity between Fenichel's persistence theorem and Darboux invariance theorem has been also demonstrated in Ginoux (2009).
Applications of the flow curvature method
Van der Pol model
Let us consider the model of Van der Pol (1926)
\[\dot{\vec{X}} \left( \begin{array}{c} \dot{x} \\ \dot{y} \\ \end{array} \right) = \vec{\Im} \left( \begin{array}{c} f \left( x,y \right) \\ g \left( x,y \right) \\ \end{array} \right) = \left( \begin{array}{c} \frac{1}{\varepsilon} \left( x+y-\frac{x^3}{3} \right) \\ -x \\ \end{array} \right)\]
By using Geometric Singular Perturbation Theory the slow invariant manifold equation associated with this system may be written as a regular perturbation expansion up to a suitable order \(O ( \varepsilon^k )\ :\)
\[y=Y \left(x,\varepsilon \right) = Y_0 (x) + \varepsilon Y_1 (x)+ \varepsilon^2 Y_2 (x)+ O ( \varepsilon^3 )\]
where functions \(Y_i(x)\) with \(i=0, \ldots, k\) may be computed:
\[ y = Y \left( x,\varepsilon \right) = \frac{x^3}{3}-x + \varepsilon \frac{x}{1-x^2}+ \varepsilon^2 \frac{x \left(1+x^2 \right)}{\left( 1 - x^2 \right)^4}+ O ( \varepsilon^3 ) \]
According to Proposition 1 the slow invariant manifold equation associated with this system reads:
\[\phi ( \vec{X} ) = \mbox{ det} ( \dot{\vec{X}}, \ddot{\vec{X}} )= 0 \quad \Leftrightarrow \quad \phi ( x,y,\varepsilon ) = 9y^2+ ( 9x+3x^3 )y + 6x^4-2x^6+9x^2 \varepsilon =0\]
Moreover, according to Proposition 2 (Darboux invariance theorem) the slow manifold equation associated with this system is locally invariant.
Since in this case the slow invariant manifold equation is given by an implicit function it must be transformed into an explicit function by using the regular expansion written above and by identifying order by order. It leads to:
\[ y = \frac{x^3}{3}-x + \varepsilon \frac{x}{1-x^2} + \varepsilon^2 \frac{x}{\left( 1 - x^2 \right)^3}+ O ( \varepsilon^3 ) \]
So, as expected by Professor Eric Benoît (personal communication), both slow manifolds are completely identical up to order one in \(\varepsilon\ .\) At order \(\varepsilon^2\) a difference appears which is due to the fact that the first flow curvature manifold associated with a two-dimensional dynamical system is defined by the second order tensor of curvature (first curvature), i.e. by a determinant involving the first and second time derivatives of the velocity vector field. If one makes the same computation as previously but with the Lie derivative of the flow curvature manifold which is defined by a determinant containing the first and third time derivatives of the velocity vector field (third order tensor), then there is no more difference between order two in \(\varepsilon\) and functions \(Y_2(x)\) given by both methods are exactly the same. This result had been already stated by Professor Bruno Rossetto (1986) by using successive approximation method.
Lorenz model
Then main difference between Geometric Singular Perturbation Theory and Flow Curvature Method is that the former which is based on regular expansions including a small multiplicative parameter deals generally with low-dimensional two and three singularly perturbed systems while the latter can be used for any kind of n-dimensional dynamical system singularly perturbed or not since it does not require any regular expansions but only involves time derivatives of the velocity vector field. So, as an example let us focus on the Lorenz model (Lorenz, 1963) which is not singularly perturbed but considered as a slow fast dynamical system and for which the Geometric Singular Perturbation Theory fails to provide its slow manifold.
\[\dot{\vec{X}} \left( \begin{array}{c} \dot{x} \\ \dot{y} \\ \dot{z} \end{array} \right) = \vec{\Im} \left( \begin{array}{l} f \left( x,y,z \right) \\ g \left( x,y,z \right) \\ h \left( x,y,z \right) \end{array} \right) = \left( \begin{array}{c} \sigma (y - x) \\ -xz + rx - y \\ xy - bz \end{array} \right) \]
with the following set of parameters \(\sigma = 10\ ,\) \(b = \frac{8}{3}\) and by setting \(r = 28\ .\)
According to Proposition 1, the slow invariant manifold equation associated with this system reads:
\[ \phi ( \vec{X} ) = \mbox{ det} ( \dot{\vec{X}}, \ddot{\vec{X}}, \stackrel{...}{\vec{X}} ) = 0 \]
Moreover, according to Proposition 2 (Darboux invariance theorem) the slow manifold equation associated with this system is locally invariant.
Remark. Let us note that flow curvature manifold exhibits the symmetry of Lorenz model, i.e. \(\phi(-x,-y, z) = \phi(x,y,z)\ .\) Slow invariant manifold equations are available on author's website [1].
Forced Van der Pol model
In this section it will be shown that Flow Curvature Method may be extended to non-autonomous dynamical systems. As an example let us consider the forced Van der Pol model studied by Guckenheimer et al. (2002). A simple variable changes may transform this non-autonomous system into a into an autonomous one as follows:
\[ \dot{\vec{X}} \left( \begin{array}{c} \varepsilon \dot{x} \\ \dot{y} \\ \dot{\theta} \end{array} \right) = \vec{\Im} \left( \begin{array}{c} f \left( x,y,z \right) \\ g \left( x,y,z \right) \\ h \left( x,y,z \right) \end{array} \right) = \left( \begin{array}{c} x+y-\frac{x^3}{3} \\ -x+ a \sin \left( 2\pi \theta \right) \\ \omega \end{array} \right) \]
As a result, the dimension of the system is increased of one but the velocity vector field still contains a trigonometric function. In order to remove such a function another variable changes which consists in considering that a sine or a cosine function is the solution of an harmonic oscillator may transform this non-autonomous system into an autonomous one but will increase the dimension of two. The forced Van der Pol may thus be written as:
\[ \dot{\vec{X}} \left( \begin{array}{c} \dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \\ \dot{x}_4 \end{array} \right) = \vec{\Im} \left( \begin{array}{l} f_1 \left( x_1 ,x_2 ,x_3,x_4 \right) \\ f_2 \left( x_1 ,x_2 ,x_3,x_4 \right) \\ f_3 \left( x_1 ,x_2 ,x_3,x_4 \right) \\ f_4 \left( x_1 ,x_2 ,x_3,x_4 \right) \end{array} \right) = \left( \begin{array}{c} \frac{1}{\varepsilon } \left( x_1 +x_2 - \frac{x_2^3}{3} \right) \\ -x_1 +ax_3 \\ \Omega x_4 \\ -\Omega x_3 \end{array} \right) \]
where \(\varepsilon =0.002\ ,\) \(a=1.8\ ,\) \(\omega =1.342043\) and \(\Omega =2\pi \omega\ .\) Although this transformation increases the dimension of the system the Flow Curvature Method enables, according to Proposition 1, to directly compute the analytical equation associated with the slow manifold of the system which reads:
\[ \phi ( \vec{X} )= \mbox{ det} ( \dot{\vec{X}}, \ddot{\vec{X}}, \stackrel{...}{\vec{X}}, \stackrel{....}{\vec{X}} ) = 0 \]
Moreover, according to Darboux invariance theorem, i.e Proposition 2 the slow manifold of forced Van der Pol model is locally invariant.
Remark. Slow invariant manifold equations are available on author's website [2].
References
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- Coddington, E. A. & Levinson, N. (1955). Theory of Ordinary Differential Equations, Mac Graw Hill, New York.
- Cole, J. D. (1968). Perturbation Methods in Applied Mathematics, Blaisdell, Waltham, MA.
- Darboux, G. (1878). Sur les équations différentielles algébriques du premier ordre et du premier degré, Bull. Sci. Math., Sr. 2 (2), pp. 60--96, pp. 123--143, pp. 151--200.
- Fenichel, N. (1971). Persistence and Smoothness of Invariant Manifolds for Flows, Ind. Univ. Math. J., 21, pp. 193--226.
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- Ginoux, J. M., Rossetto, B. & L. O. Chua (2008). Slow Invariant Manifolds as Curvature of the flow of Dynamical Systems, Int. J. Bifurcation & Chaos, 18 (11), pp. 3409-3430.
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- Hirsch, M., Pugh, C. & Shub, M. (1977). Invariant manifolds, Lecture Notes in Math., 583, Springer-Verlag, New York.
- Jones, C.K.R.T. (1994). Geometric singular perturbation theory. Lecture Notes in Mathematics, 1609, pp. 44--118.
- Kaper, T. J. (1999). An introduction to geometric methods and dynamical systems theory for singular perturbation problems, inAnalyzing Multiscale Phenomena Using Singular Perturbation Methods, Proc. Symp. Appl. Math., 56, R. E. O'Malley, Jr., and J. Cronin, eds., Am. Math. Soc., Providence, RI, pp. 85--132.
- Levinson, N. (1950). Perturbations of discontinuous solutions of non-linear Systems of differential equations, Acta Mathematica, 82, pp. 71--106.
- Llibre, J. (2004). Integrability of polynomial differential systems, in Handbook of Differential Equations (Ordinary Differential Equations Volume I), pp.437--532. Elsevier, Northholland, 2004.
- Lorenz, E. N. (1963). Deterministic non-periodic flows, J. Atmos. Sc., 20, pp. 130--141.
- O'Malley, R. E. (1974). Introduction to Singular Perturbations, Academic Press, New York.
- O'Malley, R. E. (1991). Singular Perturbation Methods for Ordinary Differential Equations, Springer-Verlag, New York.
- Rossetto, B. (1986). Trajectoires lentes des systèmes dynamiques, Proceedings of the 7th International Conference on Analysis and Optimization of Systems, Antibes, Juan-les-Pins, France, Springer Verlag, 83, pp. 630--645, 25-27 juin 1986.
- Schlomiuk, D. (1993). Elementary first integrals of differential equations and invariant algebraic curves, Expositiones Mathematicae, 11, pp. 433--454.
- Tikhonov, N. (1948). On the dependence of solutions of differential equations on a small parameter, Mat. Sb. N.S., 22 (2), pp. 193--204 (In Russian).
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- Van der Pol, B. (1926). On 'Relaxation-Oscillations', Phil. Mag., Ser. 7, 2, pp. 978--992.
- Wasow, W. R. (1965). Asymptotic Expansions for Ordinary Differential Equations, Wiley-Interscience, New York.
- Wiggins, S. (1990). Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York.
External links
- For examples of slow manifolds associated with dynamical systems of dimension 2, 3, 4 & 5 see author's website [3]
See also
Singularly perturbed systems, Relaxation oscillator, Bifurcations
