Upper bounds are obtained for the Hausdorff dimension of compact invariant sets of ordinary differential equations which are periodic in the independent variable. From these are derived sufficient conditions for dissipative analytic n-dimensional ω-periodic differential equations to have only a finite number of ω-periodic solutions. For autonomous equations the same conditions ensure that each bounded semi-orbit converges to a critical point. These results yield some information about the Lorenz equation and the forced Duffing equation.

Some applications of Hausdorff dimension inequalities for ordinary differential equations

Periodic solutions of Duffing's equations with superquadratic potential

We survey the recent progress made in the study of harmonic, subharmonic and other solutions of the forced pendulum equation u ′′ + cu ′ + asin u = h(t) when the forcing term h is periodic, almost periodic or bounded. The results depend upon various methods of nonlinear functional analysis, critical point theory and dynamical systems.

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Seventy-five years of global analysis around the forced pendulum equation ∗

In this paper, the second order non-linear differential equation x ¨ + a ( t ) f ( x , x ˙ ) x ˙ + b ( t ) g ( x ) = p ( t , x , x ˙ ) is considered, and Lyapunov's second method is used to show that uniform boundedness and convergence to zero of all solutions of this equation together with their derivatives of the first order.

On the asymptotic behavior of solutions of certain second-order differential equations

Abstract The 1-covering property of omega limit sets is established for monotone and uniformly stable skew-product semiflows with the componentwise separating property of bounded and ordered full orbits. Then these results are applied to study the asymptotic almost periodicity of solutions to almost periodic reaction-diffusion equations and differential systems with time delays. The earlier convergence results for autonomous and periodic monotone systems are generalized to the almost periodic case without the strong monotonicity assumption.

Convergence in monotone and uniformly stable skew-product semiflows with applications

The number of periodic solutions of 2-dimensional periodic systems

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Almost periodic gross-substitute dynamical systems

Even and periodic solutions of the equation ü + g(u) = e(t)

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Ultimate boundedness of solutions for a generalized lienard equation with forcing term

Publisher Summary This chapter discusses index theorems and bifucations in Duffing's equations. It illustrates that the Duffing's equation has several periodic solutions for appropriate k and B, which consist of harmonic solutions and subharmonic solutions. Index theorems are useful to the study of bifucations of periodic solutions. The bifucation of second order subharmonic solutions arises from the existence of an inversely unstable harmonic solution and the bifucation of harmonic solutions arises from the existence of a directly unstable harmonic solution.

Fumio Nakajima

Papers

The number of periodic solutions of 2-dimensional periodic systems

Almost periodic gross-substitute dynamical systems

Even and periodic solutions of the equation ü + g(u) = e(t)

Ultimate boundedness of solutions for a generalized lienard equation with forcing term

Index Theorems and Bifucations in Duffing's Equations