Bifurcations and limit cycles due to self‐excitation in nonlinear systems with joint friction: Initialization of isolated solution branches via homotopy methods

The main objective of this contribution is finding isolated stationary solutions, e.g. equilibria or limit cycles, in dynamical systems. Usually, NEWTON‐type methods are applied for solving the resulting algebraic equation system. Here, the most difficult point is finding adequate initial conditions that are providing a solution on the isolated branch. So, there is a need for a more straight forward manner of initialising the continuation of isolated solutions. Within this contribution, homotopy methods are applied. The crucial point is to define a simplified version of the problem F(x; λ), which can be continuously transformed into the original one. As an example, limit cycles of a friction oscillator including COULOMB damping is discussed and two types of homotopy maps are addressed to obtain a starting point for their continuation.


Motivation
Isolated solution branches can occur in various types of dynamical systems. For instance, buckling problems considering an imperfection can involve isolated branches of static equilibria. Fig. 1 shows the degenerated bifurcation diagram of a buckling column as a function of a load p, where an imperfection ||ψ 0 || > 0 gives rise to an isolated branch. In this case, an analytical solution is available and every solution is directly obtained [1]. However, for general problems closed form solutions will usually not be available.
So, there is a need for more generally procedures to calculate these isolated domains. The most evident way would be choosing a similar problem G(x; µ) = 0, which has a known or obvious solution. Then, a continuous transformation shall lead from the simplified problem to the original one, F (x; µ) = 0. Finally, a solution of the original problem may be obtained, and can be traced using classical path following techniques.

Homotopy methods
When applying NEWTON's method, finding adequate initial conditions may be a crucial point. To guarantee better convergence and expand the basin of attraction of a solution x, homotopy methods can be applied. The general idea is to define a map where a simplified or cheap problem G(x; µ) = 0 is continuously transformed into the complex or expensive problem F (x; µ) holding the bifurcation parameter µ constant. Within this contribution, for designing the homotopy map H λ , a convex approach was selected, in the form of a weighted linear superposition related to the homotopy parameter λ. Following the path γ = (x , λ) , where H λ (γ) = 0 and λ ∈ [0, 1] holds, a solution of the expensive problem F (x; µ) = 0 is obtained. Here, one has to assume, the expensive problem has a solution, γ has finite length and the implicit function theorem holds [2].

Selection of the cheap problem
To start the continuation of γ in a efficient way, the choice of a suitable G is reasonable. Here, a trivial determination of the cheap problems solution would be feasible. First, the global homotopy, where holds, provides a trivial solution x = a at λ = 0 [2]. To make it even more simple, the probability-one homotopy was established [2]. Here, the cheap problem provides at λ = 0 only one solution x = a. The nomenclature results from the fact, that it can be mathematically shown that the smooth path γ converges to a solution with the probability of one [2].

Results
In the following, periodic limit cycles of a friction oscillator including COULOMB damping [3], are approximated using the shows the continuation related to homotopy parameter λ for the two different homotopy maps H λ (x; µ) shown in section 2.1, while µ = −0.5 is kept constant. The path γ = γ(s) is parametrized by the arc-length s and a classical predictor-corrector scheme is used. The starting point a is the solution of the underlying smooth system for µ = −0.5.