In this paper we show how analysis from numerical computation of orbits can be applied to prove the existence of snapback repellers in discrete dynamical systems. That is, we present a computer-assisted method to prove the existence of a snapback repeller of a specific map. The existence of a snapback repeller of a dynamical system implies that it has chaotic behavior [F. R. Marotto, J. Math. Anal. Appl. 63, 199 (1978)]. The method is applied to the logistic map and the discrete predator-prey system.

In this paper we consider the following problemwhere λ > 0 is a parameter. We assume lim |x|→∞ f (x, u) =f (u) uniformly on any compact subset of [0, ∞), and do not require f (x, u) f (u) for all x ∈ R N . We prove that there exists +∞ > λ * > 0 such that ( ) has exactly two positive solutions for λ ∈ (0, λ * ), no solution for λ > λ * , a unique solution for λ = λ * , (λ * , u * ) is a turning point in C 2,α (R N ) ∩ W 2,2 (R N ), and further analyses of the set of positive solutions are made.

In this paper we present a computer-assisted technique that allows us to prove rigorously that a transversal homoclinic orbit of discrete dynamical systems with this method is simpler than previous works on this subject.

<p style='text-indent:20px;'>We consider a conical body facing a supersonic stream of air at a uniform velocity. When the opening angle of the obstacle cone is small, the conical shock wave is attached to the vertex. Under the assumption of self-similarity for irrotational motions, the Euler system is transformed into the nonlinear ODE system. We reformulate the problem in a non-dimensional form and analyze the corresponding ODE system. The initial data is given on the obstacle cone and the solution is integrated until the Rankine-Hugoniot condition is satisfied on the shock cone. By applying the fundamental theory of ODE systems and technical estimates, we construct supersonic solutions and also show that no matter how small the opening angle is, a smooth transonic solution always exists as long as the speed of the incoming flow is suitably chosen for this given angle.</p>

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