Hartman–Grobman Theorem
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, in the study of
dynamical systems In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
, the Hartman–Grobman theorem or linearisation theorem is a theorem about the local behaviour of dynamical systems in the
neighbourhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neighbourh ...
of a
hyperbolic equilibrium point In the study of dynamical systems, a hyperbolic equilibrium point or hyperbolic fixed point is a fixed point that does not have any center manifolds. Near a hyperbolic point the orbits of a two-dimensional, non-dissipative system resemble hyperbol ...
. It asserts that
linearisation In mathematics, linearization (British English: linearisation) is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the stu ...
—a natural simplification of the system—is effective in predicting qualitative patterns of behaviour. The theorem owes its name to
Philip Hartman Philip Hartman (May 16, 1915 – August 28, 2015) was an American mathematician at Johns Hopkins University The Johns Hopkins University (often abbreviated as Johns Hopkins, Hopkins, or JHU) is a private university, private research u ...
and David M. Grobman. The theorem states that the behaviour of a dynamical system in a domain near a hyperbolic equilibrium point is qualitatively the same as the behaviour of its
linearization In mathematics, linearization (British English: linearisation) is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the ...
near this equilibrium point, where hyperbolicity means that no eigenvalue of the linearization has real part equal to zero. Therefore, when dealing with such dynamical systems one can use the simpler linearization of the system to analyse its behaviour around equilibria.


Main theorem

Consider a system evolving in time with state u(t)\in\mathbb R^n that satisfies the differential equation du/dt=f(u) for some
smooth map In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; t ...
f \colon \mathbb^n \to \mathbb^n. Now suppose the map has a hyperbolic equilibrium state u^*\in\mathbb R^n: that is, f(u^*)=0 and the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of component ...
A= partial f_i/\partial x_j/math> of f at state u^* has no
eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
with real part equal to zero. Then there exists a neighbourhood N of the equilibrium u^* and a
homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
h \colon N \to \mathbb^n, such that h(u^*)=0 and such that in the neighbourhood N the flow of du/dt=f(u) is topologically conjugate by the continuous map U=h(u) to the flow of its linearisation dU/dt=AU. A like result holds for iterated maps, and for fixed points of flows or maps on manifolds. A mere topological conjugacy does not provide geometric information about the behavior near the equilibrium. Indeed, neighborhoods of any two equilibria are topologically conjugate so long as the dimensions of the contracting directions (negative eigenvalues) match and the dimensions of the expanding directions (positive eigenvalues) match. But the topological conjugacy in this context does provide the full geometric picture. In effect, the nonlinear phase portrait near the equilibrium is a thumbnail of the phase portrait of the linearized system. This is the meaning of the following regularity results, and it is illustrated by the saddle equilibrium in the example below. Even for infinitely differentiable maps f, the homeomorphism h need not to be smooth, nor even locally Lipschitz. However, it turns out to be
Hölder continuous Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition * Hölder's inequality In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality (mathematics), inequality between Lebesgue integration, in ...
, with exponent arbitrarily close to 1. Moreover, on a surface, i.e., in dimension 2, the linearizing homeomorphism and its inverse are continuously differentiable (with, as in the example below, the differential at the equilibrium being the identity) but need not be C^2. And in any dimension, if f has Hölder continuous derivative, then the linearizing homeomorphism is differentiable at the equilibrium and its differential at the equilibrium is the identity. The Hartman–Grobman theorem has been extended to infinite-dimensional Banach spaces, non-autonomous systems du/dt=f(u,t) (potentially stochastic), and to cater for the topological differences that occur when there are eigenvalues with zero or near-zero real-part.


Example

The algebra necessary for this example is easily carried out by a web service that computes normal form coordinate transforms of systems of differential equations, autonomous or non-autonomous, deterministic or
stochastic Stochastic (; ) is the property of being well-described by a random probability distribution. ''Stochasticity'' and ''randomness'' are technically distinct concepts: the former refers to a modeling approach, while the latter describes phenomena; i ...
. Consider the 2D system in variables u = (y,z) evolving according to the pair of coupled differential equations \frac = -3y + yz \quad\text\quad \frac = z+y^2. By direct computation it can be seen that the only equilibrium of this system lies at the origin, that is u^* = 0. The coordinate transform, u = h^(U) where U = (Y,Z), given by \begin y & \approx Y + YZ + \tfrac Y^3 + \tfrac Y Z^2 \\ ptz & \approx Z - \tfrac Y^2 - \tfrac Y^2 Z \end is a smooth map between the original u = (y,z) and new U = (Y,Z) coordinates, at least near the equilibrium at the origin. In the new coordinates the dynamical system transforms to its linearisation \frac = -3Y \quad\text\quad \frac = Z. That is, a distorted version of the linearisation gives the original dynamics in some finite neighbourhood.


See also

*
Linear approximation In mathematics, a linear approximation is an approximation of a general function (mathematics), function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order ...
* Stable manifold theorem


References


Further reading

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External links

* * * {{DEFAULTSORT:Hartman-Grobman Theorem Theorems in mathematical analysis Theorems in dynamical systems Approximations