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On the stability of periodic orbits for differential systems in \(\mathbb{R}^n\). (English) Zbl 1169.34039

The authors consider a differential system
\[ \frac{dx}{dt}=X(x), \]
where \(X: U\subset \mathbb R^n\rightarrow \mathbb R^n\) is a \(C^1\) function and assume that the system exhibits a periodic orbit \(\Gamma=\{\gamma(t)\mid 0\leq t<T\}\). The goal of the paper is to compute the characteristic multipliers associated with the periodic orbit which determine its stability. The traditional approach is to use the first order variational equation
\[ \frac{du}{dt}=DX(\gamma(t))u(t),\quad u(0)=\text{Id}. \]
In this paper, the authors offer a new method, assuming that the periodic orbit is given by the transversal intersection of \(n-1\) codimension one hypersurfaces. The main theorem is
Theorem 1.1. Let \(\Gamma=\{\gamma(t)\mid 0\leq t<T\}\) be a \(T\)-periodic solution. Consider a smooth function \(f: U\subset \mathbb R^n\rightarrow \mathbb R^{n-1}\) such that
1.
\(\Gamma\) is contained in \(\bigcap_{i=1,\dots,n-1}\{f_i(x)=0\}\),
2.
the crossings of all manifolds \(\{f_i(x)=0\}\) are transversal over \(\Gamma\),
3.
there exists a \((n-1)\times (n-1)\) matrix \(k(x)\) satisfying \[ Df(x)X(x)=k(x)f(x). \]
Let \(v(t)\) be the \((n-1)\times (n-1)\) fundamental matrix solution of
\[ \frac{dv}{dt}=k(\gamma(t))v(t),\;v(0)=\text{Id}. \]
Then the characteristic multipliers of \(\Gamma\) are the eigenvalues of \(v(T)\).
The two-dimensional version of this theorem was given in [H. Giacomini and M. Grau, J. Differ. Equations 213, No. 2, 368–388 (2005; Zbl 1074.34034)]. The authors apply this method to three examples: Besides a \(4\)- and a \(3\)-dimensional polynomial problem, where the latter is related to Mathieu’s equation, the authors study a \(6\)-dimensional system modeling the dynamics of a rigid body. V. A. Steklov [“New particular solution of differential equations of motion of a heavy rigid body about a fixed point”, Trudy Ob-va estest. 1, 1–3 (1899)] found a periodic orbit of this system, and due to three independent first integrals at least 4 characteristic multipliers are equal to one. In this paper, it is shown that the remaining two can be studied using a linear second order differential equation.

MSC:

34D08 Characteristic and Lyapunov exponents of ordinary differential equations
37D05 Dynamical systems with hyperbolic orbits and sets
34C25 Periodic solutions to ordinary differential equations
70E50 Stability problems in rigid body dynamics

Citations:

Zbl 1074.34034
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