Reduction Methods and Chaos for Quadratic Systems of Differential Equations

Abstract

We consider systems of differential equations with quadratic nonlinearities having applications for biochemistry and population dynamics, which may have a large dimension n. Due to the complexity of these systems, reduction algorithms play a crucial role in study of their large time behavior. Our approach aims to reduce a large system to a smaller one consisting of m differential equations, where m << n. Under some restrictions (that allow us to separate slow and fast variables in the system) we obtain a new system of differential equations, involving slow variables only. This reduction is feasible from a computational point of view for large n that allows us to investigate sensitivity of dynamics with respect to random variations of parameters. We show that the quadratic systems are capable to generate all kinds of structurally stable dynamics including chaos.

Complex bifurcations in Bénard–Marangoni convection

“…The key point (iii) allows us to exhibit the existence of chaotic dynamics. This is found in [8], where it is shown that all kinds of structurally stable dynamics can be generated by quadratic systems. Thus, all structurally stable dynamics can appear as a result of bifurcations.…”

“…Proof. According to results [8] for any ǫ 1 > 0 we can construct ǫ 1 -realization of the field Q by semiflows defined by (6.119). Moreover, due to the previous lemma, for any ǫ 2 > 0 we can find ǫ 2 -realization of any system (6.119) by semiflows defined by (6.98).…”

“…At the bifurcation point, by standard methods of invariant manifold theory [12,13,21], we derive a system of differential equations with quadratic nonlinearities for mode magnitudes X j (t), j = 1, ..., N, that is reminiscent of the Lorenz system. Previous results [8,11] show that the corresponding dynamical system can have periodic or chaotic trajectories.…”

Solving System of Higher-Order Linear Differential Equations on the Level of Operators

“…where EIE −1 is the Green's operator of the first order system. Now, the solution of the given IVP ( 4) is obtained by collecting the first n rows ofũ as in equation (5). The uniqueness of the solution follows from the fact that homogeneous IVP has the trivial solution.…”

Solving System of Higher-Order Linear Differential Equations on the Level of Operators

“…al. [5] considered a large system of quadratic differential equations and presented a reduction algorithm to reduce the large system to smaller systems under some restrictions. Mukesh [6] presented a new technique to solve higher order ordinary differential equations using modified Adomian decomposition method.…”