MATHEMATICS

T. SABIROV

ON THE STABILITY OF SMALL PERIODIC SOLUTIONS AT MULTIDIMENSIONAL DEGENERACIES

(Presented by Academician A. Yu. Ishlinskii on 20 XI 1967)

1. Consider the system of ordinary differential equations

\[ dx/dt=f(t,x,\lambda) \tag{1} \]

with right-hand sides that are \(\omega\)-periodic in \(t\) and depend on the scalar parameter \(\lambda\). We shall assume that the right-hand sides possess sufficient smoothness with respect to the aggregate of the variables.

Throughout the paper it is assumed that system (1), for all \(\lambda\), has the zero solution: \(f(t,0,\lambda)\equiv 0\). We shall be interested in the question of the existence and stability of small nonzero \(\omega\)-periodic solutions of system (1) (see \((^1,^2)\)).

We shall call a number \(\lambda_0\) (see \((^3,^4)\)) a bifurcation value of the parameter \(\lambda\) if to every \(\varepsilon>0\) there corresponds a number \(\lambda\in(\lambda_0-\varepsilon,\lambda_0+\varepsilon)\) for which system (1) has a nonzero \(\omega\)-periodic solution whose amplitude \(\max \|x(t)\|\) does not exceed \(\varepsilon\).

Denote by \(V(t,\lambda)\) the fundamental matrix (see \((^5)\)) of the system linearized at zero,

\[ dx/dt=A(t,\lambda)x. \tag{2} \]

It is easy to see (see \((^3,^5)\)) that only those values of the parameter \(\lambda\) for which the monodromy matrix \(V(\omega,\lambda)\) has the eigenvalue 1 can be bifurcation values. The converse assertion is not true in general. We note that a number of important results on bifurcation points were obtained in \((^6)\).

For simplicity we shall assume that 1 is an eigenvalue of the matrix \(V(\omega,0)\), and we shall be interested in the question of the existence and stability of small periodic solutions for small values of \(\lambda\).

The case when 1 is a simple eigenvalue of the matrix \(V(\omega,0)\) was studied in detail in \((^3)\). In \((^7)\) the case was studied in which the eigenvalue 1 has order equal to 1 and its multiplicity is \(k\). In this note the general case is considered.

1. Let \(e_j^i\) \((i=1,\ldots,k;\ j=0,\ldots,l_i-1)\) be the eigenvectors and associated vectors of the matrix \(V(\omega,0)\) corresponding to the eigenvalue 1, i.e.

\[ V(\omega,0)e_0^i=e_0^i \quad (i=1,\ldots,k);\qquad V(\omega,0)e_j^i=e_j^i+e_{j-1}^i. \]

Let \(g_j^i\) be the eigenvectors and associated vectors of the adjoint matrix, i.e.

\[ V^*(\omega,0)g_0^i=g_0^i \quad (i=1,\ldots,k);\qquad V^*(\omega,0)g_j^i=g_j^i+g_{j-1}^i \quad (j=1,\ldots,l_i-1). \]

Without loss of generality (see \((^8,^9)\)) one may assume that the equalities

\[ (e_i^j,g_p^q)=1 \quad \text{when } j=q,\ i+p=l_j-1; \]

\[ (e_i^j,g_p^q)=0 \quad \text{when } j\ne q \text{ or } i+p\ne l_j-1. \tag{3} \]

Write system (1) in the form

\[ dx/dt=A(t,\lambda)x+B_m[t,x,\lambda]+F(t,x,\lambda), \tag{4} \]

where \(B_m[t,x,\lambda]\) denotes the homogeneous terms of order \(m\) in the spatial variables \(x\), and \(F(t,x,\lambda)\) denotes terms of higher order of smallness.

Let \(x_{1,0}(t),\ldots,x_{k,0}(t)\) denote solutions of the linear system (2) for \(\lambda=0\), satisfying the initial conditions \(x_{i,0}(0)=e_0^i\) \((i=1,\ldots,k)\). Introduce the homogeneous operator of degree \(m\)

\[ \Phi_m[\eta]=\left\{\int_0^\omega \left(V^{-1}(\tau,0)B_m\left[\tau;\sum_{j=1}^k \eta_j x_{j,0}(\tau);0\right],g_0^i\right)d\tau\right\}_{i=1,\ldots,k}, \tag{5} \]

where \(\eta=(\eta_1,\ldots,\eta_k)\).

It can be shown that, in the case when there exists an \(\alpha>0\) for which the inequality

\[ \|\Phi_m[\eta]\|\geq \alpha\|\eta\|^m, \tag{6} \]

is satisfied, system (1) for \(\lambda=0\) has no small nonzero \(\omega\)-periodic solutions.

Next write the matrix \(A(t,\lambda)\) in the form

\[ A(t,\lambda)=A_0(t)+\lambda^p A_1(t)+o(\lambda^p), \tag{7} \]

where \(p>0\) is some integer. Let

\[ B_0=\left(\int_0^\omega \left(V^{-1}(\tau,0)A_1(\tau)x_{i,0}(\tau),g_0^j\right)d\tau\right)_1^k . \]

Then the following is true.

Theorem 1. Suppose condition (6) is satisfied and the equation

\[ \Phi_m[\sigma]+B_0\sigma=0 \tag{8} \]

has a finite number of real solutions \(\sigma_1,\ldots,\sigma_q\). Suppose, moreover, that each matrix
\(m\Phi_m[\sigma_i^{m-1};\,\cdot]+B_0\) \((i=1,\ldots,q)\) is nonsingular.

Then zero is a bifurcation value of the parameter, and for every sufficiently small value \(\lambda>0\) system (1) has \(q\) small nonzero \(\omega\)-periodic solutions.

The proof uses the well-known Schmidt transformation \((^{10,11})\) and the results of work \((^{12})\).

Equation (8) will be called the characteristic equation. If the matrices
\(m\Phi_m[\sigma_i^{m-1};\,\cdot]+B_0\) are singular, then the roots of the characteristic equation split the small \(\omega\)-periodic solutions of system (1) into groups, i.e., for each root \(\sigma_i\) of the characteristic equation one can determine a finite or infinite number of small nonzero \(\omega\)-periodic solutions that depend “essentially” on \(\sigma_i\). The group of small nonzero \(\omega\)-periodic solutions corresponding to the root \(\sigma_i\) will be denoted by \(K(\sigma_i)\). In connection with these assertions we also note work \((^{13})\).

1. It is said that to the eigenvalue \(\lambda_0\) of the matrix \(A_0\) there corresponds a Jordan block of multiplicity \(k\) and length \(l\), if to the zero eigenvalue of the matrix \(A_0-\lambda I\) there correspond \(k\) Jordan blocks of equal length \(l\) (see \((^8)\)).

Let \(\sigma_i\) be some root of the characteristic equation. Introduce the matrix
\(\Delta_i=B_0-m\Phi_m[\sigma_i^{m-1};\,\cdot]\).

Theorem 2. Suppose the monodromy matrix \(V(\omega,0)\) has the eigenvalue \(1\), to which there corresponds one Jordan block of multiplicity \(k\) and length \(l\).

Then, if the matrix \(\Delta_i\) has at least one nonzero eigenvalue, then for \(l \geq 3\) all small \(\omega\)-periodic solutions from the group \(K(\sigma_i)\) are Lyapunov unstable.

Denote by \(S\) the linear span of the eigenvectors and generalized eigenvectors of the matrix \(V(\omega,0)\) corresponding to the eigenvalue \(1\). Suppose that to \(1\) there correspond \(r\) Jordan boxes of lengths \(l_1,\ldots,l_r\) and multiplicities \(q_1,\ldots,q_r\), respectively. Denote the linear span of the \(q_j\) eigenvectors and the corresponding generalized eigenvectors by \(S_j\). There is a decomposition \(S=S_1\oplus\cdots\oplus S_r\), and

\[ \dim S=\sum_{i=1}^{r} q_i l_i,\qquad k=\sum_{i=1}^{r} q_i . \]

Denote by \(S_j^0\) the linear span of the eigenvectors in \(S_j\), so that \(\dim S_j^0=q_j\). Denote the projector onto \(S_j^0\) by \(P_j\).

Theorem 3. Suppose the matrix \(V(\omega,0)\) has the eigenvalue \(1\), to which there correspond \(r\) Jordan boxes of multiplicities \(q_1,\ldots,q_r\) and lengths \(l_1,\ldots,l_r\), respectively.

Then, if for some \(j_0\) the matrix \(P_{j_0}\Delta_iP_{j_0}\) has a nonzero eigenvalue, then for \(l_{j_0}\geq 3\) all nonzero small \(\omega\)-periodic solutions of the group \(K(\sigma_j)\) are Lyapunov unstable.

In the case when \(l_j\leq 2\) \((j=1,\ldots,r)\), the appearance of both stable and unstable small \(\omega\)-periodic solutions in the group \(K(\sigma_j)\) is possible. The corresponding criteria are rather cumbersome. We do not give them here.

The author expresses sincere gratitude to M. A. Krasnosel’skii for constant advice and attention to the results of this note.

Voronezh State
University

Received
1 XI 1967

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