# A fixed point approach for decaying solutions of difference equations

Zuzana Došlá, Mauro Marini and Serena Matucci

( )

Abstract. A boundary value problem associated to the difference equation with
advanced argument

$\Delta\bigl{(}a_{n}\Phi(\Delta x_{n})\bigr{)}+b_{n}\Phi(x_{n+p})=0,\ \ n\geq
1$ ($*$)

is presented, where $\Phi(u)=|u|^{\alpha}$sgn $u,$ $\alpha>0,p$ is a positive
integer and the sequences $a,b,$ are positive. We deal with a particular type
of decaying solutions of ($*$ ‣ A fixed point approach for decaying solutions
of difference equations), that is the so-called intermediate solutions (see
below for the definition) . In particular, we prove the existence of these
type of solutions for ($*$ ‣ A fixed point approach for decaying solutions of
difference equations) by reducing it to a suitable boundary value problem
associated to a difference equation without deviating argument. Our approach
is based on a fixed point result for difference equations, which originates
from existing ones stated in the continuous case. Some examples and
suggestions for future researches complete the paper.

Keywords. Nonlinear difference equation, functional discrete equations,
boundary value problem on the half line, fixed point theorem, decaying
solution.

MSC 2010: Primary 39A22 , Secondary 47H10

##  1 Introduction

Consider the equation

$\Delta\bigl{(}a_{n}\Phi(\Delta x_{n})\bigr{)}+b_{n}\Phi(x_{n+p})=0,$ (P)

where $\Delta$ is the forward difference operator $\Delta
x_{n}=x_{n+1}-x_{n},$ $\Phi$ is the operator
$\Phi(u)=|u|^{\alpha}\operatorname{sgn}u$, $\alpha>0$, $p$ is a positive
integer and the sequences $a,b$ are positive for $n\geq 1$, and satisfy

$\sum_{i=1}^{\infty}\Phi^{\ast}\left(\frac{1}{a_{i}}\right)<\infty,\quad\sum_{i=1}^{\infty}b_{i}=\infty,$
(1)

where $\Phi^{\ast}$ is the inverse of the map $\Phi,$ that is
$\Phi^{\ast}(u)=|u|^{1/\alpha}$sgn $u.$

Equation (P) appears in the discretization process for searching spherically
symmetric solutions of certain nonlinear elliptic equations with weighted
$\varphi$-Laplacian, see, e.g., [9]. A special case of (P) is the discrete
half-linear equation

$\Delta\bigl{(}a_{n}\Phi(\Delta y_{n})\bigr{)}+b_{n}\Phi(y_{n+1})=0,$ (H)

which has been studied extensively from various points of view, especially
with regard to the oscillation and the qualitative behavior of nonoscillatory
solutions, see [12, Chapter 3] and references therein.

As usual, a nontrivial solution $x$ of (P) is said to be nonoscillatory if
$x_{n}$ is either positive or negative for any large $n$ and oscillatory
otherwise. In virtue of the Sturm separation criterion, see, e.g. [12, Section
8.2.1.], all the solutions of (H) have the same behavior with respect to the
oscillation. In other words, either all nontrivial solutions of (H) are
nonoscillatory or all the solutions of (H) are oscillatory. Thus,
equivalently, we say that (H) is nonoscillatory if (H) has a nonoscillatory
solution. Nevertheless, for equation (P) with $p\neq 1,$ the situation is
different, since in this case oscillatory solutions and nonoscillatory
solutions may coexist.

In this paper we deal with the existence of a particular type of
nonoscillatory solutions, that is solutions $x$ of (P) such that
$x_{n}>0,\Delta x_{n}<0$ for large $n$ and

$\lim_{n}x_{n}=0,\ \ \lim_{n}x_{n}^{[1]}=a_{n}\Phi(\Delta x_{n})=-\infty,$ (2)

where $x^{[1]}$ is called the quasidifference of $x.$ Solutions satisfying (2)
are called intermediate solutions. This terminology originates from the
corresponding continuous case and it is due to the fact that, when (1) is
satisfied, then any nonoscillatory solution $x$ of (P) satisfies either
$\lim_{n}x_{n}=\ell_{x}\neq 0,$ or (2), or
$\lim_{n}x_{n}=0,\lim_{n}x_{n}^{[1]}=-\ell_{x},0<\ell_{x}<\infty,$ see [5, 7].
The investigation of intermediate solutions is an hard problem, due to
difficulties in finding suitable sharp upper and lower bounds for these
solutions, see, e.g., [2, page 241] and [14, page 3], in which these facts are
pointed out for the continuous case.

In the half-linear case the problem of existence of intermediate solutions has
been completely solved by the following.

######  Theorem 1.

Assume (1).

$(i_{1})$ Equation (H) does not have intermediate solutions if

$\sum_{n=1}^{\infty}b_{n}\Phi\left(\sum_{k=n+1}^{\infty}\Phi^{\ast}\left(\frac{1}{a_{k}}\right)\right)+\sum_{n=1}^{\infty}\Phi^{\ast}\left(\frac{1}{a_{n+1}}\sum_{k=1}^{n}b_{k}\right)\,<\infty\,.$

$(i_{2})$ Equation (H) has intermediate solutions if (H) is nonoscillatory and

$\sum_{n=1}^{\infty}b_{n}\Phi\left(\sum_{k=n+1}^{\infty}\Phi^{\ast}\left(\frac{1}{a_{k}}\right)\right)+\sum_{n=1}^{\infty}\Phi^{\ast}\left(\frac{1}{a_{n+1}}\sum_{k=1}^{n}b_{k}\right)\,=\infty.$

Proof.  The assertion follows from Theorem 3.1 (b), (c) and Theorem 3.2. in
[6], with minor changes. $\Box$

Here, we present a comparison result which allows to solve the boundary value
problem [BVP]

$\begin{cases}\Delta\bigl{(}a_{n}\Phi(\Delta
x_{n})\bigr{)}+b_{n}\Phi(x_{n+p})=0,&p>1,\\\ \lim_{n}x_{n}=0,\
\lim_{n}x_{n}^{[1]}=-\infty&\end{cases}$ (3)

by reducing it to the existence of intermediate solutions in the half-linear
case. The main result is the following.

######  Theorem 2.

Assume (1) and

$\limsup_{n}b_{n}<\infty.$ (4)

Then the BVP (3) is solvable, i.e. equation (P) with $p>1$ has intermediate
solutions, if and only if the half-linear equation

$\Delta\bigl{(}a_{n+p-1}\Phi(\Delta y_{n})\bigr{)}+b_{n}\Phi(y_{n+1})=0,$ (H1)

has intermediate solutions.

Theorem 2 will be proved by means of a fixed point result for discrete
operators acting in Fréchet spaces, see Theorem 4 below.

Combining Theorem 1 and the comparison Theorem 2, we get necessary and
sufficient conditions for existence of intermediate solutions of difference
equation with advanced argument.

######  Corollary 1.

Assume (1) and (4).

$(i_{1})$ Equation (P) with $p>1$ has intermediate solutions if (H1) is
nonoscillatory and

$\sum_{n=1}^{\infty}b_{n}\Phi\left(\sum_{k=n+1}^{\infty}\Phi^{\ast}\left(\frac{1}{a_{k+p-1}}\right)\right)+\sum_{n=1}^{\infty}\Phi^{\ast}\left(\frac{1}{a_{n+p}}\sum_{k=1}^{n}b_{k}\right)\,=\infty\,.$
(5)

(i${}_{2})$ Equation (P) with $p>1$ does not have intermediate solutions if

$\sum_{n=1}^{\infty}b_{n}\Phi\left(\sum_{k=n+1}^{\infty}\Phi^{\ast}\left(\frac{1}{a_{k+p-1}}\right)\right)+\sum_{n=1}^{\infty}\Phi^{\ast}\left(\frac{1}{a_{n+p}}\sum_{k=1}^{n}b_{k}\right)\,<\infty\,.$

##  2 Fixed point approaches

Boundary value problems for difference equations in $\mathbb{R}^{n}$ are often
solved by reducing the problem to a fixed point equation for a possibly
nonlinear operator in a suitable function space. Thus, the existence of a
solution is obtained by applying a fixed point theorem, for instance the
Tychonoff theorem, the Schauder theorem, the Leray-Schauder continuation
principle or Krasnoselkii types fixed point theorems on cones. For a survey on
this topic we refer to the papers [1, 3, 15] and the monographies [4, 17]. In
particular, in [17, Chapter 2], see also [4, Chapter 5], certain BVPs are
studied by means of a nonlinear Leray-Schauder alternative. This approach is
based on a very general method given in [13]. In particular, in [13] the
authors present a Leray-Schauder continuation principle in locally convex
topological vector spaces, which unifies the Leray-Schauder alternative
theorem and the Tichonov fixed point theorem. More precisely, let $\mathbb{E}$
be a Hausdorff locally convex topological vector space with a family of
seminorms generating the topology. The following holds.

######  Theorem 3.

[13, Theorem 1.1] Let $Q$ be a convex closed subset of $\mathbb{E}$ and let
$T:Q\times[0,1]\rightarrow\mathbb{E}$ be a continuous map with relatively
compact image. Assume that:

i${}_{1})$ $T(x,0)\in Q$ for any $x\in Q;$

i${}_{2})$ for any $(x,\lambda)\in\partial Q\times[0,1)$ with $T(x,\lambda)=x$
there exists open neighborhoods $U_{x}$ of $x$ in $\mathbb{E}$ and
$I_{\lambda}$ of $\lambda$ in $[0,1)$ such that

$T\Bigl{(}(U_{x}\cap\partial Q)\times I_{\lambda}\Bigr{)}\subset Q.$

Then the equation

$x=T(x,1)$

has a solution.

Some of the above quoted results have a discrete counterpart. For instance, a
method for solving BVPs associated to difference systems is given in [15,
Theorem 2.1]. Due to the peculiarities of the discrete case, it may be applied
to functional difference equations, including equations with deviating
arguments or sum difference equations.

Now, we present an existence result which generalizes, in the particular case
of scalar difference equations, [15, Theorem 2.1].

Denote by $\mathbb{N}_{n}$ and $\mathbb{N}_{m,n}$, the sets

$\mathbb{N}_{n}=\\{i\in\mathbb{N}:\,i\geq n\in\mathbb{N}\\}$
$\mathbb{N}_{m,n}=\\{i\in\mathbb{N}_{m}:\,i<n,\,m,n\in\mathbb{N},m<n\\}$

and let $\mathbb{X}$ be the space of all real sequences defined on
$\mathbb{N}_{m}$. Hence $\mathbb{X}$ is a Frechét space with the topology of
pointwise convergence on $\mathbb{N}_{m}$. From the discrete Arzelà-Ascoli
theorem (see, e.g., [4, Theorem 5.3.1]), any bounded set in $\mathbb{X}$ is
relatively compact. We recall that a set $\Omega\subset\mathbb{X}$ is bounded
if and only if it consists of sequences which are equibounded on
$\mathbb{N}_{m,n}$ for any $n>m.$ Clearly, if $\Omega\subset$ $\mathbb{X}$ is
bounded, then $\Omega^{\Delta}=\\{\Delta u,\,u\in\Omega\\}$ is bounded, too.

Using, with minor changes, a discrete counterpart of a compactness and
continuity result stated in [8, Theorem 1.3] for the continuous case, we have
the following.

######  Theorem 4.

Consider the BVP

$\begin{cases}\Delta(a_{n}\Phi(\Delta x_{n}))=g(n,x),&n\in\mathbb{N}_{m}\\\
x\in S,&\end{cases}$ (6)

where $g:\mathbb{N}_{m}\times\mathbb{X}\rightarrow\mathbb{R}$ is a continuous
map, and $S$ is a subset of $\ \mathbb{X}$.
Let $G:\,\mathbb{N}_{m}\times\mathbb{X}^{2}\rightarrow\mathbb{R}$ be a
continuous map such that $G(k,u,u)=g(k,u)$ for all
$(k,u)\in\mathbb{N}_{m}\times\mathbb{X}$. If there exist a nonempty, closed,
convex set $\Omega\subset\mathbb{X}$, and a bounded, closed subset
$S_{C}\subset S\cap\Omega$ such that the problem

$\begin{cases}\Delta(a_{n}\Phi(\Delta x_{n}))=G(n,x,q),&n\in\mathbb{N}_{m}\\\
x\in S_{C},&\end{cases}$ (7)

has a unique solution for any $q\in\Omega$ fixed, then (6) has at least a
solution.

Proof.  Let $T$ be the operator $T:\Omega\rightarrow S_{C}$ which maps every
$q\in\Omega$ into the unique solution $x=T(q)$ of (7). Let us show that the
operator $T$ is continuous with relatively compact image. The relatively
compactness of $T(\Omega)$ follows immediately since $S_{C}$ is bounded. To
prove the continuity of $T$ in $\Omega$, let $\\{q^{j}\\}$ be a sequence in
$\Omega$, ${q}^{j}\rightarrow{q}^{\infty}\in\Omega$, and let
${v}^{j}=T(q^{j})$. Since $T(\Omega)$ is relatively compact, $\\{{v}^{j}\\}$
admits a subsequence (still indicated with $\\{{v}^{j}\\}$) which converges to
${v}^{\infty}\in\mathbb{X}$. As $v^{j}\in S_{C}$ and $S_{C}$ is closed, then
${v}^{\infty}\in S_{C}$. Taking into account the continuity of $G$, we obtain

$\Delta(a_{n}\Phi(\Delta v_{k}^{\infty}))=\lim_{j}\Delta(a_{n}\Phi(\Delta
v_{k}^{j}))=\lim_{j}G(k,q^{j},v^{j})=G(k,q^{\infty},{v}^{\infty}).$

The uniqueness of the solution of (7) yields $v^{\infty}=T({q}^{\infty})$, and
therefore $T$ is continuous on $\Omega$. By the Tychonoff fixed point theorem,
$T$ has at least one fixed point in $\Omega$, which is a solution of (6), as
it can be easily checked, taking into account that $S_{C}\subset S$. $\Box$

As follows from the proof of Theorem 4, no explicit form of the fixed point
operator is needed for the solvability of (6). A key point for the unique
solvability of (7) in Theorem 4 is the choice of the map $G$. To this aim, in
our opinion, the best cases are the following two, namely

$i_{1})\text{ \ }G(n,q,x)=\tilde{g}(n,q),\text{ \ \ \ }i_{2})\text{ \ \
}G(n,q,x)=\tilde{g}(n,q)\Phi(x_{n+1}).$

In the case $i_{2}),$ the equation in (7) is a half-linear equation and in
this situation a very large variety of results is known, see, e.g., [12,
Chapter VIII]. An application in this direction is in [15, Section 4].

In the case $i_{1}),$ that is when the function $G$ does not depend on $x$ and
the equation in (7) is affine, Theorem 4 can be particularly useful to solve
BVPs associated to difference equations with deviating arguments. Indeed, in
this case, it can lead to a BVP associated to a second order difference
equation without deviating argument. An application of this fact is in the
following section.

##  3 Proof of Theorem 2

For proving Theorem 2, the following auxiliary result is needed.

######  Lemma 1.

Assume (1) and (4). Let $x$ be an eventually positive decreasing solution of
(P) such that $\lim_{n}x_{n}=0$. Then the series

$\sum_{i=2}^{\infty}\Phi^{\ast}\left(\frac{1}{a_{i+p-1}}\sum_{k=1}^{i-1}b_{k}\Phi(x_{k+p})\right)$

converges.

Proof.  Without loss of generality, suppose $0<x_{n}\leq 1,$ $\Delta x_{n}<0$
for $n\geq m_{0}\geq 1$. We claim that for any $k,j\geq m_{0}$ we have

$\left|x_{j}^{[1]}-x_{k}^{[1]}\right|\leq B|j-k|,$ (8)

where $B=\sup_{i\geq 1}b_{i}.$ For simplicity, suppose $k\geq j$. Summing (P)
we have

$x_{k}^{[1]}-x_{j}^{[1]}=-\sum_{i=j}^{k-1}b_{i}\Phi(x_{i+p}),$ (9)

where, as usual, $\sum_{i=k_{1}}^{k}\gamma_{i}=0$ if $k<k_{1}$. Since
$\left|x_{i}\right|\leq 1$ and $b_{i}\leq B,$ the inequality (8) follows.
Further, from (9),

$\displaystyle\sum_{i=m_{0}}^{\infty}\Phi^{\ast}\left(\frac{1}{a_{i+p-1}}\sum_{k=m_{0}}^{i-1}b_{k}\Phi(x_{k+p})\right)=\sum_{i=m_{0}}^{\infty}\Phi^{\ast}\left(\frac{1}{a_{i+p-1}}\left(x_{m_{0}}^{[1]}-x_{i}^{[1]}\right)\right)=$
$\displaystyle=\sum_{i=m_{0}}^{\infty}\Phi^{\ast}\left(\frac{1}{a_{i+p-1}}\left(x_{m_{0}}^{[1]}-x_{i}^{[1]}-x_{i+p-1}^{[1]}+x_{i+p-1}^{[1]}\right)\right).$

Since

$\left|x_{i}^{[1]}-x_{i+p-1}^{[1]}+x_{i+p-1}^{[1]}-x_{m_{0}}^{[1]}\right|\leq\left|x_{i}^{[1]}-x_{i+p-1}^{[1]}\right|+\left|x_{i+p-1}^{[1]}\right|+\left|x_{m_{0}}^{[1]}\right|,$

in view of (8) we obtain

$\displaystyle\left|\sum_{i=m_{0}}^{\infty}\Phi^{\ast}\left(\frac{1}{a_{i+p-1}}\left(x_{i}^{[1]}-x_{m_{0}}^{[1]}\right)\right)\right|\leq$
$\displaystyle\leq\sum_{i=m_{0}}^{\infty}\Phi^{\ast}\left(\frac{B(p-1)+\left|x_{m_{0}}^{[1]}\right|}{a_{i+p-1}}+\frac{\left|x_{i+p-1}^{[1]}\right|}{a_{i+p-1}}\right).$

In virtue of (1), the series

$\sum_{i=m_{0}}^{\infty}\Phi^{\ast}\left(\frac{B(p-1)+\left|x_{m_{0}}^{[1]}\right|}{a_{i+p-1}}\right)$

converges. Since

$\sum_{i=m_{0}}^{\infty}\Phi^{\ast}\left(\frac{\left|x_{i+p-1}^{[1]}\right|}{a_{i+p-1}}\right)=-\sum_{i=m_{0}}^{\infty}\Delta
x_{i+p-1}=x_{m_{0}+p-1},$

using (1) and the inequality

$\Phi^{\ast}(X+Y)\leq\sigma_{\alpha}(\Phi^{\ast}(X)+\Phi^{\ast}(Y)),$

where

$\sigma_{\alpha}=\left\\{\begin{array}[c]{ccc}1&&\text{if }\alpha\geq 1\\\
2^{(1-\alpha)/\alpha}&&\text{if }\alpha<1\end{array}\right.,$

we obtain the assertion. $\Box$

Proof of Theorem 2. First, we prove that if (P) has intermediate solutions,
then (H1) has intermediate solutions.

Let $x$ be an intermediate solution of (P) and, without loss of generality,
assume for $n\geq n_{0}\geq 1$

$0<x_{n}<1,\text{ \ \ }\Delta x_{n}<0\text{ }.$ (10)

In view of (4), there exists $L>0$ such that for any $n\geq n_{0}$

$\sum_{i=n}^{n+p-2}b_{i}\leq L.$ (11)

Moreover, let $M$ be a positive constant, $M<1,$ such that

$\Phi(M)\leq\frac{\left|x_{n_{0}}^{[1]}\right|}{L+\left|x_{n_{0}}^{[1]}\right|}.$
(12)

Let $\mathbb{X}$ be the Fréchet space of real sequences defined for $n\geq
n_{0}$, endowed with the topology of convergence on $\mathbb{N}_{n_{0}}$, and
consider the subset $\Omega\subset$ $\mathbb{X}$ defined by

$\Omega=\big{\\{}u\in\mathbb{X}\,:Mx_{n+p-1}\leq u_{n}\leq
x_{n+p-1}\big{\\}}.$

For any $u\in\Omega$ consider the boundary value problem [BVP]

$\begin{cases}\Delta\bigl{(}a_{n+p-1}\Phi(\Delta
z_{n})\bigr{)}+b_{n}\Phi(u_{n+1})=0,&n\geq n_{0}\\\
z_{n_{0}}^{[1]}=x_{n_{0}}^{[1]},\ \lim_{n}z_{n}=0,&\end{cases}$ (13)

where $z^{[1]}$ denotes the quasidifference of $z,$ that is

$z_{n}^{[1]}=a_{n+p-1}\Phi(\Delta z_{n}).$ (14)

For any $u\in\Omega$ we have

$\sum_{k=n_{0}}^{n}b_{k}\Phi(u_{k+1})\leq\sum_{k=n_{0}}^{n}b_{k}\Phi(x_{k+p}).$

Hence, using Lemma 1, we have

$\lim_{n}\sum_{i=n}^{\infty}\Phi^{\ast}\left(\frac{1}{a_{i+p-1}}\left(\sum_{k=n_{0}}^{i-1}b_{k}\Phi(u_{k+1})\right)\right)=0.$

Thus, a standard calculation shows that for any $u\in\Omega$ the BVP (13) has
the unique solution $z$. Let $\mathcal{T}$ be the operator which associates to
any $u\in\Omega$ the unique solution $z$ of (13).

Summing the equation in (13) and using (9) we get

$z_{n}^{[1]}=x_{n_{0}}^{[1]}-\sum_{k=n_{0}}^{n-1}b_{k}\Phi(u_{k+1})\geq
x_{n_{0}}^{[1]}-\sum_{k=n_{0}}^{n-1}b_{k}\Phi(x_{k+p})=x_{n}^{[1]}.$

Since $x^{[1]}$ is decreasing for $n\geq n_{0}$ and $p>1,$ we obtain for
$n\geq n_{0}$

$z_{n}^{[1]}\geq x_{n+p-1}^{[1]}$ (15)

i.e., in view of (14),

$a_{n+p-1}\Phi(\Delta z_{n})\geq a_{n+p-1}\Phi(\Delta x_{n+p-1}),$

that is,

$\Delta z_{n}\geq\Delta x_{n+p-1}.$

Since $\lim_{i}z_{i}=\lim_{i}x_{i}=0,$ we obtain for $n\geq n_{0}$

$z_{n}\leq x_{n+p-1}.$ (16)

Now, let us prove that for $n\geq n_{0}$

$z_{n}\geq Mx_{n+p-1}.$ (17)

Summing the equation in (13) and using (14) we get

$z_{n}^{[1]}=x_{n_{0}}^{[1]}-\sum_{k=n_{0}}^{n-1}b_{k}\Phi(u_{k+1})\leq
x_{n_{0}}^{[1]}-\Phi(M)\sum_{k=n_{0}}^{n-1}b_{k}\Phi(x_{k+p}),$

or, using (9),

$\displaystyle z_{n}^{[1]}$ $\displaystyle\leq
x_{n_{0}}^{[1]}+\Phi(M)\left(x_{n}^{[1]}-x_{n_{0}}^{[1]}\right)=$ (18)
$\displaystyle=\Phi(M)x_{n+p-1}^{[1]}+\Phi(M)\left(x_{n}^{[1]}-x_{n+p-1}^{[1]}\right)+(1-\Phi(M))x_{n_{0}}^{[1]}.$

From (9), (10) and (11), we also have

$x_{n}^{[1]}-x_{n+p-1}^{[1]}=\sum_{i=n}^{n+p-2}b_{i}\Phi(x_{i+p})\leq\sum_{i=n}^{n+p-2}b_{i}\leq
L.$

Thus, from (18) we obtain

$z_{n}^{[1]}\leq\text{
}\Phi(M)x_{n+p-1}^{[1]}+\left(L+\left|x_{n_{0}}^{[1]}\right|\right)\Phi(M)+x_{n_{0}}^{[1]}.$
(19)

In view of (12) we have

$\left(L+\left|x_{n_{0}}^{[1]}\right|\right)\Phi(M)+x_{n_{0}}^{[1]}\leq 0.$

Hence, from (19) we get for $n\geq n_{0}$

$z_{n}^{[1]}\leq\Phi(M)x_{n+p-1}^{[1]}$ (20)

or, in view of (14), $\Delta z_{n}\leq M\text{ }\Delta x_{n+p-1},$ and (17)
follows, since $\lim_{i}z_{i}=\lim_{i}x_{i}=0.$ Thus, in virtue of (16) and
(17), the operator $\mathcal{T}$ maps $\Omega$ into itself, that is

$\mathcal{T}(\Omega)\subset\Omega.$

Denote by $S$ the boundary conditions in (13), i.e.

$S=\left\\{v\in\mathbb{X}:\,a_{n_{0}+p-1}\Phi(\Delta
v_{n_{0}})=x_{n_{0}}^{[1]},\ \lim_{n}v_{n}=0\right\\}$

For any $z\in\mathcal{T}(\Omega)$ we have $z\in S$. Since
$\mathcal{T}(\Omega)\subset\Omega$, we get $z\in\Omega\cap S$.

Denote by $S_{C}$ the subset of $\mathbb{X}$ given by

$S_{C}=S\cap\Omega.$

Since $\lim_{n}x_{n}=0$, it holds

$S_{C}=\left\\{v\in\mathbb{X}:\,a_{n_{0}+p-1}\Phi(\Delta
v_{n_{0}})=x_{n_{0}}^{[1]},\,Mx_{n+p-1}\leq v_{n}\leq x_{n+p-1}\right\\}\,.$

Thus $S_{C}$ is a bounded and closed subset of $\mathbb{X.}$ Applying Theorem
4 we obtain that the operator $\mathcal{T}$ has a fixed point $\overline{z}\in
S_{C}$. Clearly the sequence $\overline{z}$ is a solution of (H1) and
$\lim_{n}\overline{z}_{n}=0.$ Since $\overline{z}\in\mathcal{T}(\Omega),$ from
(15) and (20), we get

$x_{n+p-1}^{[1]}\leq\overline{z}_{n}^{[1]}\leq\Phi(M)x_{n+p-1}^{[1]}$

and so $\lim_{n}\overline{z}_{n}^{[1]}=-\infty.$ Hence $\overline{z}$ is an
intermediate solution of (H1).

Now, we prove the vice-versa, that is if (H1) has intermediate solutions, then
(P) has intermediate solutions.

The argument is similar to the one above given, with minor changes. Let $y$ be
an intermediate solution of (H1) such that for $n\geq n_{0}\geq 1$

$0<y_{n}<1,\text{ \ \ }\Delta y_{n}<0\text{,}$

and define

$n_{1}=n_{0}+p.$

In view of (4), there exists $\Lambda>0$ such that for any $n\geq n_{1}$

$\sum_{i=n-p+1}^{n-1}b_{i}\leq\Lambda.$ (21)

Without loss of generality, we can suppose

$\Lambda<\left|y_{n_{1}}^{[1]}\right|,$ (22)

where $y_{n_{1}}^{[1]}=a_{n_{1}+p-1}\Phi(\Delta y_{n_{1}}).$ Moreover, let
$H>1$ be a positive constant such that

$\Phi(H)\geq\frac{\left|y_{n_{1}}^{[1]}\right|}{\left|y_{n_{1}}^{[1]}\right|-\Lambda}.$
(23)

Let $\mathbb{X}_{1}$ be the Fréchet space of the real sequences defined for
$n\geq n_{1},$ endowed with the topology of convergence on
$\mathbb{N}_{n_{1}}$. Define the subset $\Omega_{1}$ of $\mathbb{X}_{1}$

$\Omega_{1}=\big{\\{}u\in\mathbb{X}_{1}\,:y_{n-p+1}\leq u_{n}\leq
Hy_{n-p+1}\big{\\}}$

and for any $u\in\Omega_{1}$ consider the BVP

$\begin{cases}\Delta\bigl{(}a_{n}\Phi(\Delta
w_{n})\bigr{)}+b_{n}\Phi(u_{n+p})=0,&n\geq n_{1}\\\
w_{n_{1}}^{[1]}=y_{n_{1}}^{[1]},\ \lim_{n}w_{n}=0,&\end{cases}$ (24)

where $w^{[1]}$ and $y^{[1]}$ denote the quasidifferences of $w$ and $y,$
respectively, that is the sequences

$w_{n}^{[1]}=a_{n}\Phi(\Delta w_{n}),\text{ \ \ \
}y_{n}^{[1]}=a_{n+p-1}\Phi(\Delta y_{n}).\text{\ }$ (25)

As before, for any $u\in\Omega_{1}$ the BVP (24) has a unique solution $w$.
Thus, let $\mathcal{T}$ be the operator which associates to any
$u\in\Omega_{1}$ the unique solution $w$ of (24). Summing the equation in (24)
and using (25) we get

$w_{n}^{[1]}=y_{n_{1}}^{[1]}-\sum_{k=n_{1}}^{n-1}b_{k}\Phi(u_{k+p})\leq
y_{n_{1}}^{[1]}-\sum_{k=n_{1}}^{n-1}b_{k}\Phi(y_{k+1})=y_{n}^{[1]}\,.$ (26)

Since $y^{[1]}$ is decreasing for $n\geq n_{0}$ and $p>1,$ we have
$y_{n}^{[1]}\leq y_{n-p+1}^{[1]}$ for $n\geq n_{1}$, and from (26) we obtain

$w_{n}^{[1]}\leq y_{n-p+1}^{[1]}$ (27)

i.e.

$\Delta w_{n}\leq\Delta y_{n-p+1},$

which implies

$w_{n}\geq y_{n-p+1},$ (28)

since $\lim_{i}w_{i}=\lim_{i}y_{i}=0$.

Now, let us prove that

$w_{n}\leq H\text{ }y_{n-p+1}\text{ \ \ }.$ (29)

Summing the equation in (24) we get

$w_{n}^{[1]}=y_{n_{1}}^{[1]}-\sum_{k=n_{1}}^{n-1}b_{k}\Phi(u_{k+p})\geq
y_{n_{1}}^{[1]}-\Phi(H)\sum_{k=n_{1}}^{n-1}b_{k}\Phi(y_{k+1}),$

that is

$\begin{split}w_{n}^{[1]}&\geq
y_{n_{1}}^{[1]}+\Phi(H)\left(y_{n}^{[1]}-y_{n_{1}}^{[1]}\right)\\\
&=\Phi(H)y_{n-p+1}^{[1]}+\left(y_{n}^{[1]}-y_{n-p+1}^{[1]}\right)\Phi(H)+(1-\Phi(H))y_{n_{1}}^{[1]}.\end{split}$
(30)

From (H1) and (21) we have

$y_{n}^{[1]}-y_{n-p+1}^{[1]}=-\sum_{i=n-p+1}^{n-1}b_{i}\Phi(y_{i+1})\geq-\Lambda.$

Thus, from (30) we obtain

$\displaystyle w_{n}^{[1]}$
$\displaystyle\geq\Phi(H)y_{n-p+1}^{[1]}-\Lambda\Phi(H)+y_{n_{1}}^{[1]}-\Phi(H)y_{n1}^{[1]}$
$\displaystyle=\Phi(H)y_{n-p+1}^{[1]}+\left(\left|y_{n_{1}}^{[1]}\right|-\Lambda\right)\Phi(H)+y_{n_{1}}^{[1]},$

or, in view of (22) and (23),

$w_{n}^{[1]}\geq\Phi(H)y_{n-p+1}^{[1]}$ (31)

i.e.,

$\Delta w_{n}\geq H\text{ }\Delta y_{n-p+1}.$

Since $\lim_{i}w_{i}=\lim_{i}y_{i}=0,$ from here we get (29). Hence, in virtue
of (28) and (29), the operator $\mathcal{T}$ maps $\Omega_{1}$ into itself,
i.e.

$\mathcal{T}(\Omega_{1})\subset\Omega_{1}.$

Using the same argument to the one given in the sufficiency part, denote by
$S_{1}$ the boundary conditions in (24). Applying Theorem 4 with
$S_{C}=S_{1}\cap\Omega_{1}$, we get that the operator $\mathcal{T}$ has a
fixed point $\overline{w}\in S_{1}\cap\Omega_{1}.$ Clearly the sequence
$\overline{w}$ is a solution of (P) and $\lim_{n}\overline{w}_{n}=0.$ Since
$\overline{w}\in\mathcal{T}(\Omega_{1})$, from (27) and (31), we get

$Hy_{n-p+1}^{[1]}\leq\overline{w}_{n}^{[1]}\leq y_{n-p+1}^{[1]}$

and so $\lim_{n}\overline{w}_{n}^{[1]}=-\infty.$

Hence $\overline{w}$ is an intermediate solution of (P) and the proof is
complete. $\Box$

##  4 Suggestions and examples

The following example illustrates Theorem 2 and Corollary 1.

Example 1. Consider the difference equation with advanced argument

$\Delta\bigl{(}(n-p+1)^{1+\alpha}\Phi(\Delta
x_{n})\bigr{)}+\gamma\Phi(x_{n+p})=0,\text{ \ \ }n\geq p\geq 2,$ (32)

where $\gamma$ is a positive constant. Using Theorem 2 and Corollary 1, it is
easy to show that (32) has intermediate solutions if and only if

$0<\gamma\leq\left(\frac{1}{1+\alpha}\right)^{\alpha+1}.$ (33)

Indeed, consider the half-linear equation

$\Delta\bigl{(}n^{1+\alpha}\Phi(\Delta x_{n})\bigr{)}+\gamma\Phi(x_{n+1})=0.$
(34)

A standard calculation shows that (1) is satisfied. Moreover, using the change
of variable

$y_{n}=n^{1+\alpha}\Phi(\Delta x_{n})$ (35)

the equation (34) is transformed into the generalized discrete Euler equation

$\Delta\bigl{(}\Phi^{\ast}(\Delta
y_{n})\bigr{)}+\gamma^{1/\alpha}\left(\frac{1}{n+1}\right)^{(1+\alpha)/\alpha}\Phi^{\ast}(y_{n+1})=0,$

which is nonoscillatory if $\gamma$ satisfies (33) and oscillatory if

$\gamma>\left(\frac{1}{1+\alpha}\right)^{\alpha+1},$ (36)

see, e.g., [18]. Since the transformation (35) maintains the oscillatory
behavior, see, e.g., [11], we get that (34) is nonoscillatory if and only if
(33) is satisfied. Moreover, as $a_{n}=n^{1+\alpha},$ $b_{n}=\gamma,$ we have

$\sum_{n=p}^{\infty}\Phi^{\ast}\left(\frac{1}{a_{n+p}}\sum_{k=p}^{n}b_{k}\right)=\gamma^{1/\alpha}\sum_{n=p}^{\infty}\left(\frac{n}{n+p}\right)^{1/\alpha}\frac{1}{n+p}=\infty$

and so the condition (5) is satisfied. Hence, from Corollary 1$-i_{1})$
equation (32) has intermediate solutions if (33) is satisfied. When (36)
holds, as claimed, the half-linear equation (34) is oscillatory and it does
not admit intermediate solutions. Thus, from Theorem 2, equation (32) does not
have intermediate solutions, too.

The existence of intermediate solutions for (P) does not depend on condition
(4), as the following example shows.

Example 2. Consider the difference equation with advanced argument

$\Delta\bigl{(}(n+1)!\text{ }\Delta x_{n}\bigr{)}+(n+p)!\text{
}x_{n+p}=0,\text{ \ }p\geq 2.$ (37)

A direct computation shows that $x=\left\\{1/n!\right\\}$ is an intermediate
solution of (37). Nevertheless, for (37), assumption (4) is not verified and
Theorem 2 cannot be applied. Hence, it is an open problem if Theorem 2
continues to hold when condition (4) failed.

Example 1 suggests the following two comparison results.

######  Corollary 2.

Assume (1) and (4). Suppose that

$a_{n}\geq n^{1+\alpha},\text{ \ \ \ and \ \
}{\displaystyle\sum\limits_{n=N}^{\infty}}\Phi^{\ast}\left(\frac{n}{a_{n+p}}\right)=\infty,$

where $N\geq p.$ Set $L=\sup_{n\geq N}b_{n}.$ If $b$ is bounded away from zero
and

$L<\left(\frac{1}{1+\alpha}\right)^{1+\alpha},$

then the equation

$\Delta\bigl{(}(n-p+1)^{1+\alpha}\Phi(\Delta
x_{n})\bigr{)}+b_{n}\Phi(x_{n+p})=0,\text{ \ \ }n\geq p\geq 2,$ (38)

has intermediate solutions.

Proof. Consider the equation

$\Delta\bigl{(}n^{1+\alpha}\Phi(\Delta x_{n})\bigr{)}+L\Phi(x_{n+1})=0.$ (39)

Reasoning as in Example 1, we get that (39) is nonoscillatory. Hence, in
virtue of the Sturm comparison theorem, see, e.g., [12, Chapter 8.2], the
half-linear equation (H) is nonoscillatory. Moreover, since $b$ is bounded
away from zero, there exists $\varepsilon>0$ such that $b_{n}\geq\varepsilon$
for any $n\geq 1.$ Hence

${\displaystyle\sum\limits_{n=N}^{\infty}}\Phi^{\ast}\left(\frac{1}{a_{n+1}}\sum_{k=1}^{n}b_{k}\right)\geq\Phi^{\ast}(\varepsilon){\displaystyle\sum\limits_{n=N}^{\infty}}\Phi^{\ast}\left(\frac{n}{a_{n+1}}\right)=\infty,$

which implies (5). Hence, applying Corollary 1$-i_{1})$ to the equation (38),
we get the assertion. $\Box$

######  Corollary 3.

Assume (1) and (4). Moreover, suppose that for $n\geq N\geq p,$

$a_{n}\leq n^{1+\alpha}.$

Set $\ell=\inf_{n\geq N}b_{n}.$ If

$\ell>\left(\frac{1}{1+\alpha}\right)^{1+\alpha},$

then the equation (38) does not have intermediate solutions.

Proof. The argument is similar to the one given in Corollary 2. Consider the
half-linear equation

$\Delta\bigl{(}n^{1+\alpha}\Phi(\Delta x_{n})\bigr{)}+\ell\text{
}\Phi(x_{n+1})=0.$ (40)

Reasoning as before, we get that (40) is oscillatory. Hence, in virtue of the
Sturm comparison theorem, also (H) is oscillatory, and so (H) does not have
intermediate solutions. Thus, applying Theorem 2 to the equation (H), we
obtain the assertion. $\Box$

Some suggestions for future researches are in order.

1) As claimed, Theorem 3 from [13] gives a very general fixed point result,
which is based on a continuation principle in a Hausdorff locally convex
space. Further, in [13] the solvability to certain BVPs in the continuous case
is also given. It should be interesting to establish corresponding discrete
versions of these existence results, especially for [13, Theorem 2.2], which
deals with a scalar equation. It could be useful for studying discrete BVPs
when it is hard to find an appropriate bounded closed set $\Omega$ which is
mapped into itself, as, for instance occurs for intermediate solutions to
Emden-Fowler superlinear discrete equation

$\Delta(a_{n}|\Delta x_{n}|^{\alpha}\operatorname{sgn}\Delta
x_{n})+b_{n}|x_{n+1}|^{\beta}\operatorname{sgn}x_{n+1}=0\,,\text{ \ \
}\alpha<\beta,$

see, e.g., [6, Section V.].

2) The proof of Theorem 2 does not work if $p\leq 0.$ Indeed, in this case the
half-linear equation (H1) is not defined, due to the shift in the weight
coefficient $a$ of the discrete operator

$(Dx)_{n}=\Delta(a_{n}\Phi(\Delta x_{n})).$

Further, when $p<0,$ the sequence $\left\\{u_{n+p}\right\\}$ in BVP (24) has
to be defined not only for $n\geq n_{1},$ but also for any $i\geq n_{1}+p.$
Consequently, when (P) is an equation with delay, the solvability of (3)
requires a different approach, which will be presented in a forthcoming paper
[10].

3) Theorem 2 establishes a comparison between the asymptotic decay of
intermediate solutions of (P) with the one of an associated half-linear
equation. Recently, in some particular cases, a precise asymptotic analysis of
intermediate solutions for discrete half-linear equations has been made in the
framework of regular variation, see [16]. It should be interesting to apply
this approach for obtaining a precise description of the asymptotic behavior
also for intermediate solutions of the equations with advanced argument.

## References

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_Authors’ addresses_ :

_Zuzana Došlá,_ Department of Mathematics and Statistics, Masaryk University,
Kotlářská 2, CZ-61137 Brno, Czech Republic.
E-mail<EMAIL_ADDRESS>
_Mauro Marini,_ Department of Mathematics and Computer Science ”Ulisse Dini”,
University of Florence, Via di S. Marta 3, I-50139 Florence, Italy.
E-mail<EMAIL_ADDRESS>
_Serena Matucci,_ Department of Mathematics and Computer Science ”Ulisse
Dini”, University of Florence, Via di S. Marta 3, I-50139 Florence, Italy.
E-mail<EMAIL_ADDRESS>


$\omega>\omega_{p}$. Hence, we conclude that suitable conditions to observe
the propagation of the shear-polariton in $Z(\omega)$ are
$\omega\ll\omega_{p}$ and $\omega>v_{F}^{*}\left|q\right|$, as described in
Sec. VIII.2.3.

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††thanks<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Fractal Patterns in the Parameter Space of Bi-stable Duffing Oscillator

Md Nahid Hasan  Department of Mechanical Engineering, University of Utah, Salt
Lake City, UT 84112, USA  Taylor E. Greenwood  Department of Mechanical
Engineering, University of Utah, Salt Lake City, UT 84112, USA  Robert G.
Parker  Department of Mechanical Engineering, University of Utah, Salt Lake
City, UT 84112, USA  Yong Lin Kong  Department of Mechanical Engineering,
University of Utah, Salt Lake City, UT 84112, USA  Pai Wang  Department of
Mechanical Engineering, University of Utah, Salt Lake City, UT 84112, USA

###### Abstract

We study the dissipative bi-stable Duffing oscillator with equal energy wells
and observe fractal patterns in the parameter space of driving frequency,
forcing amplitude, and damping ratio. Our numerical investigation reveals the
Hausdorff fractal dimension of the boundaries that separate the oscillator’s
intra-well and inter-well behaviors. Furthermore, we categorize the inter-well
behaviors as three steady-state types: switching, reverting, and vacillating.
While fractal patterns in the phase space are well-known and heavily studied,
our results point to a new research direction about fractal patterns in the
parameter space. Another implication of this study is that the vibration of a
continuous bi-stable system modeled using a single-mode approximation also
manifests fractal patterns in the parameter space. In addition, our findings
can guide the design of next-generation bi-stable and multi-stable mechanical
metamaterials.

††preprint: APS/123-QED

Introduction: The dissipative bi-stable Duffing oscillator is a well-known
dynamical system with applications in many fields, such as shape morphing [1,
2, 3, 4, 5, 6, 7, 8, 9, 10], energy harvesting [11, 12, 13, 14, 15, 16, 17,
18], soft robotics [19, 20, 21], MEMS devices [22, 23, 24, 25, 26, 27], energy
absorption [28, 29, 30], and drug delivery [31]. In addition, with recent
advances in the multi-stable metamaterial [32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44], and advanced functional systems [45, 46, 47, 48], bi-stable
system dynamics can be introduced as a mechanism for folding [49, 50] and
reprogramming [51]. As illustrated in Fig. 1(a), one physical realization of
the dissipative bi-stability consists of two linear elastic connections of
equal spring constants, one linear damping, and a point mass. The system has
two stable equilibria and one unstable equilibrium. The generalized governing
equation is:

$\begin{split}m\frac{d^{2}\hat{u}}{dt^{2}}+c\frac{d\hat{u}}{dt}-k_{1}\hat{u}+k_{3}\hat{u}^{3}=F\cos(\omega
t),\end{split}$ (1)

which contains a cubic non-linearity term emerging from the geometrical
arrangement shown in Fig. 1(a). After non-dimensionalization, Eq. (1) leads to
the governing equation of the classical symmetric bi-stable Duffing oscillator
[52, 53, 54, 55, 7],

$\ddot{u}+\gamma\dot{u}-u+\mu u^{3}=G\cos(\Omega\tau),$ (2)

where $\ddot{u}={d^{2}u}/{d\tau^{2}}$, $\dot{u}={du}/{d\tau}$, and the non-
dimensional displacement and time are defined as $u=\hat{u}/{l_{c}}$ and
$\tau=t\omega_{n}$ for the natural frequency $\omega_{n}=\sqrt{{k_{1}}/{m}}$
at the linear limit. Other dimensionless parameters are
$\gamma={c}/({m\omega_{n}})$, $\mu={k_{3}{l_{c}}^{2}}/({m\omega_{n}^{2}})$,
$\Omega={\omega}/{\omega_{n}}$, and $G={F}/({l_{c}\omega_{n}^{2}})$. For
$\mu=1$, Eq. (2) results in the symmetric energy potential shown in Fig. 1(b).
If we choose the characteristic length in such a way that
$l_{c}=\sqrt{{k_{1}}/{k_{3}}}\;\Rightarrow\;\mu=1$, the potential landscape
has double symmetric wells around two stable equilibria at $u_{-1}=-1$ and
$u_{+1}=+1$, which are separated by an unstable hilltop equilibrium at
$u_{0}=0$ (see Supplemental Materials [56] for detailed derivation).

Figure 1:  (a) Physical setup that behaves as a dissipative bi-stable Duffing
oscillator. Here, the zero-force spring length $L_{o}>b$ makes the central
configuration (the gray-shaded state in the middle) unstable. (b) Double-well
potential with an energy barrier at the center with $\mu=1$. (c) Five
frequency response curves across values of forcing amplitude $G$. (d) Forcing
amplitude-frequency ($G$ vs. $\Omega$) parameter space where magenta and
yellow zones represent inter-well and intra-well behavior regions,
respectively. (e) Time response curves show the inter-well (magenta) and
intra-well (yellow) behaviors. (f) Phase spaces of inter-well (magenta) and
intra-well (yellow) behaviors.

The frequency response curve [57, 58, 59, 60, 61, 62] is a standard tool to
describe the bi-stable system dynamics by solving Eq. (2) in the frequency
domain, but it does not provide a complete picture. Instead, analyses on the
potential well escape provide advantages to investigate the overall dynamics
[63, 64, 65, 66, 67]. For example, energy criteria [68, 69, 70, 71, 72, 73],
forcing phase [74], and velocity conditions [75] offer valuable information.
However, due to the uncertainty of the initial conditions [76, 77] and fractal
basin boundaries in the phase space [78, 79, 80, 81, 82, 83, 84, 85, 86, 75,
87], most recent studies are unable to conclusively determine in which
potential well (right or left) the system will reside after the escape. In
addition, system parameters such as frequency ($\Omega$), forcing amplitude
($G$), and damping ratio ($\gamma$) also influence the system’s final state. A
previous publication [88] analyzed the forcing amplitude-frequency parameter
plane and reported the fractal boundary between the inter-well and intra-well
[52] behaviors. A fractal dimension of 1.26 was calculated from scarce
experimental data. More robust data analyses are necessary to understand the
fractal boundary’s statistical complexity. Several other studies also reported
regions of various shapes [89, 90, 91, 92] and the Arnold tongue [93] in the
parameter space of the Duffing system. However, none of them investigated the
fractal nature of the region boundaries.

In this study, we propose numerical criteria addressing the oscillator’s
categories of behavior between potential wells. To avoid the well-known
sensitivity due to the initial condition, we fix the initial conditions at
$(u,\dot{u})=(-1,0)$ in all simulations. Furthermore, we seek to conduct
rigorous numerical simulations to categorize different system behaviors and
calculate accurate fractal dimensions in the parameter space.

Figure 2:  Categories of time-domain and phase-space of behaviors with damping
ratio, $\gamma=0.07$: (a,b) Switching behavior with parameters
$(G,\Omega)=(0.100,1.17595)$. (c,d) Reverting behavior with parameters
$(G,\Omega)=(0.125,1.209412)$. (e,f) Periodic vacillating behavior with
parameters $(G,\Omega)=(0.400,1.10294)$. (g,h) Aperiodic vacillating behavior
with parameters $(G,\Omega)=(0.175,1.13291)$. (i,j) Intra-well behavior with
parameters $(G,\Omega)=(0.150,1.30506)$

.

We first re-examine the frequency response curve and the parameter space of
the bi-stable Duffing oscillator. Importantly, our numerical investigation
reveals that the single-degree-of-freedom (SDOF) model based on single-
vibration-mode assumption exhibits the fractal pattern very well. Next, we
categorize the different steady-state response behaviors inside the parameter
space. Finally, we calculate the fractal dimensions of the boundaries that
separate the inter-well and intra-well behaviors in the parameter space. The
fractal dimension reveals the statistical complexity of the parameter space of
the bi-stable Duffing system. Our results indicate that inter-well behavior,
traditionally defined as chaotic behavior, can be understood and has potential
applications in shape-morphing.

Frequency response curve and its limitations: We start with time-domain
simulations using the fourth-order Runge-Kutta scheme over a broad range of
($G,\Omega$) for a constant damping ratio of $\gamma=0.07$ to discern the
inter-well and intra-well responses. In Fig. 1(c), we sweep the forcing
frequency $\Omega$ in the range of $0.80\leq\Omega\leq 2.0$ at the step size
of $0.0047$. We also sweep the forcing amplitude $G$ in the range of
$0.100\leq G\leq 0.200$ at the step size of $0.025$. In each simulation, we
solve Eq. (2) to obtain the time response of 500 forcing cycles. Then we
capture the average peak-to-peak steady-state amplitude of the last 50 periods
of the time response. In all cases, we start with the initial conditions of
$(u,\dot{u})=(-1,0)$.

Figure 3:  Forcing amplitude-frequency ($G$ vs. $\Omega$) parameter space
(sampled over a uniform grid of $256\times 256$) with: (a) Damping ratio,
$\gamma=0.001$ (b) $\gamma=0.07$, (c) Zoom-in of the black-framed portion of
Fig. 3(b), which shows the fractal nature. (d) $\gamma=0.15$. (e)
$\gamma=0.25$. (f) $\gamma=0.30$.

First, we examine the frequency response curve in Fig. 1(c). It shows multiple
solutions, both inter-well and intra-well, co-existing near $\Omega=1.0$,
i.e., the system’s resonant frequency at the linear limit. This is the root
cause of the hysteresis jump phenomena [68, 69]. Consequently, the standard
frequency response curve cannot provide enough information about potential-
well escape criteria.

We then consider the parameter space of forcing amplitude and driving
frequency ($G$ and $\Omega$) and use a $256\times 256$ grid to plot the
results in Fig. 1(d). We obtain the time history for 500 periods in each
simulation. If the response amplitude, which begins at $u_{-1}=-1$ based on
the initial condition, exceeds the hilltop equilibrium at $u_{0}=0$, then the
response exhibits the inter-well behavior. Alternatively, if the response
amplitude does not exceed the unstable equilibrium at $u_{0}$ in all 500
periods, we recognize this response as the intra-well behavior. The two
different parameter regions are illustrated in Fig. 1(d). As examples, Figs.
1(e) and (f) show the time and phase space responses of the inter-well
(magenta curves) and intra-well (yellow curves) behaviors. We observe that the
boundary between the inter-well and intra-well zones in Fig. 1(d) takes an
intricate shape. Near the boundary, the oscillator’s behavior is highly
sensitive to a subtle change in the amplitude and frequency of the excitation.
This shows that the single-mode single-degree-of-freedom (SDOF) model can
capture fractal patterns in the parameter space of a bi-stable system,
contrary to the argument that higher-order vibration modes with more degrees
of freedom are required [88].

Figure 4:  Damping ratio-forcing frequency ($\gamma$ vs. $\Omega$) parameter
space (sampled over a uniform grid of $256\times 256$) with: (a) Forcing
amplitude, $G=0.100$. (b) $G=0.125$. (c) $G=0.150$. (d) $G=0.200$. (e)
$G=0.225$. (f) $G=0.250$.

Categories of steady-state behaviors in the parameter space: Fig. 1 shows two
behaviors: intra-well oscillation near one stable equilibrium; and inter-well
between two stable equilibria. However, a thorough investigation of the
dynamics of a double-well potential system with harmonic excitation uncovers
opportunities for unique applications, as it exhibits four possible types of
behavior: (i) Switching - inter-well behavior where the oscillation begins in
one potential well, and after the transient stage, reaches its steady-state in
the other potential well, (ii) Reverting - inter-well behavior where the
oscillation begins in one potential well, reaches the other potential well
during the transient stage, and eventually returns to its original potential
well in the steady state; (iii) Vacillating - inter-well behavior where the
oscillation keeps moving between the two wells, and it never settles into
either one; and (iv) Intra-well - steady state oscillation in one potential
well only.

To quantitatively distinguish these categories of behavior using numerical
results from time-domain simulations, we first check whether the system
displacement amplitude, which begins at $u_{-1}$ based on the initial
condition, exceeds the hilltop equilibrium at $u_{0}$. Then, we recognize the
last 50 cycles as the steady-state output of the total of 500 cycles in the
simulations. Based on this numerical assumption, we define
$\xi_{\textrm{max}}=\max(u(t)|t\in\\{\text{last 50 cycles}\\})$ and
$\xi_{\textrm{min}}=\min(u(t)|t\in\\{\text{last 50 cycles}\\})$. As
illustrated in Fig. 2(a) - (Note that a much smaller number of cycles are
shown here for illustration purposes), if $\xi_{\textrm{max}}>0$ and
$\xi_{\textrm{min}}>0$, we categorize this behavior as switching. In Fig.
2(b), the phase space plot of the same simulation illustrates the growth of
the oscillator’s limit cycle oscillations (LCOs) [94] from $u_{-1}$, and it
eventually reaches a steady-state trajectory near $u_{+1}$. This picture
resembles the stable LCOs in many studies [94, 74, 95]. Thus, up to the
numerical precision, we can conclude that the oscillator stabilizes in the
second stable state [95] and categorize this as the switching behavior. In
Fig. 2(c), the oscillator switches from the $u_{-1}$ potential well to the
$u_{+1}$ one during the transient stage. For the steady-state, we observe
$\xi_{\textrm{max}}<0$ and $\xi_{\textrm{min}}<0$, meaning it reverts back to
its initial well at $u_{-1}$ in the end. We categorize this behavior as
reverting. Correspondingly, we see a stable limit cycle around the stable
equilibrium at $u_{-1}$ in Fig. 2(d). Next, in Figs. 2(e) and (g), the
oscillator’s inter-well behavior exhibits persistent shifting between $u_{-1}$
to $u_{+1}$. The observation of $\xi_{\textrm{max}}>0$ and
$\xi_{\textrm{min}}<0$ in the steady-state implies that it continues to
oscillate between both potential wells without settling into either one, so we
can categorize these behaviors as vacillating. We also analyze the phase space
behavior to distinguish between periodic and aperiodic vacillating behaviors.
Fig. 2(f) illustrates periodic vacillating where the phase space trajectory
follows a high-energy orbit motion outside of the separatrix [94], while Fig.
2(h) shows low-energy orbits around either one or two potential wells inside
the separatrix. Finally, Fig. 2(i) displays the intra-well behavior, which
remains in the vicinity of the stable equilibrium at $u_{-1}$ without reaching
the unstable equilibrium at $u_{0}$. Fig. 2(j) shows the stable limit cycle
around $u_{-1}$ corresponding to the intra-well confinement.

Figure 5:  Fractal dimension of the boundaries in forcing amplitude-frequency
($G$ vs. $\Omega$) and damping ratio-forcing frequency ($\gamma$ vs. $\Omega$)
parameter space: (a) An example with the damping ratio of $\gamma=0.07$ (b)
Fractal boundary between inter-well and intra-well regions. (c) An example
with the forcing amplitude of $G=0.190$ (d) Fractal boundary between inter-
well and intra-well regions. (e) and (f) show the linear regression of the
box-counting algorithm data for (b) and (d) using the least square method to
calculate the Hausdorff fractal dimension. (g) and (h) show the downward trend
of the Hausdorff fractal dimension of the boundaries in parameter space as the
damping ratio and forcing amplitude increase (see Supplemental Materials [56]
for more results).

Fractal patterns in the parameter space: Using the four categories of
behaviors described in the last section, we update our understanding of the
parameter space. For each set of ($G$, $\Omega$), we run time-domain
simulations in the parameter ranges of $0.80\leq\Omega\leq 1.8$ and $0.03\leq
G\leq 0.30$ with different damping ratios. Figs. 3(a)-(f) show the results for
$\gamma=$ 0.001, 0.07, 0.15, 0.25, and 0.30, respectively (see Supplemental
Materials [56] for more results with other damping ratios). We categorize the
numerical steady state of each simulation either as switching, reverting,
vacillating, or intra-well behavior. We represent them as red, blue, green,
and yellow data points in Fig. 3, respectively. The minimally required forcing
amplitude $G_{\textrm{min}}$ for inter-well behavior rises as $\gamma$
increases. For example, Fig. 3(a) indicates the minimum amplitude required for
inter-well behavior is $G=0.04$. In contrast, in Figs. 3(b) and 3(c), it is
about $G=0.09$. In all cases, we see the well-known V-shaped “Arnold’s
tongue”, whose bottom tip represents $G_{\textrm{min}}$ for inter-well
behavior. As damping increases, the tip of Arnold’s tongue moves up and left
towards $\Omega=1$, which is the system’s resonant frequency at the linear
limit. Fig. 3(a), in particular, shows that the vacillating behavior dominates
among all inter-well behaviors in the low damping limit. This means that low
dissipation increases the likelihood of vacillating behavior.

In general, the results in Figs. 3 suggest that, compared to the quasi-static
actuation [36], utilizing harmonic excitation can significantly reduce the
force amplitude necessary to switch between stable states. For the same system
with static loading, the force required for switching between stable states is
$0.38$ (see Supplemental Materials [56]). In contrast, Figs. 3 reveals that
inter-well behavior can be achieved with much lower actuation peak force with
dynamic loading. These results highlight the importance of optimizing the
$(G,\Omega,\gamma)$ parameters in achieving the inter-well behavior of the
system. Furthermore, incorporating feedback control strategy [49] can further
improve controlled switching between stable states, leading to a much more
efficient actuation mechanism than quasi-static actuation.

To better understand the system behaviors, in Fig. 3(c), we zoom in on the
black-framed region in Fig. 3(b). Fig. 3(c) shows intricate feather-shaped
self-similar patterns and strongly suggests the existence of fractals in the
parameter space of the bi-stable Duffing oscillator. Motivated by the fractal
pattern in the forcing amplitude-frequency ($G$ vs. $\Omega$) parameter plane,
we also investigate the damping ratio-forcing frequency ($\gamma$ vs.
$\Omega$) parameter plane. Fig. 4 shows the results in the parameter range
$0.80\leq\Omega\leq 1.8$ and $0.00\leq\gamma\leq 0.35$ for $G=$ 0.1, 0.125,
0.15, 0.2, 0.225, and 0.25, respectively (see Supplemental Materials [56] for
more results with other forcing amplitudes). The patterns look similar to
those shown in Fig. 3, but they are flipped upside down with the top tip
indicating the maximally allowed damping for inter-well behavior. Like Fig. 3,
with increasing forcing amplitude $G$, the “Arnold’s tongue” tip moves up and
also left towards $\Omega=1$, the resonant frequency of the system at the
linear limit. Feather-shaped fractal patterns with self-similarity also
appear. The results show that higher forcing amplitude and lower dissipation
lead to a higher likelihood of vacillating behavior in the steady state.

Fractal dimensions of boundaries in parameter space: To understand the system
behavior’s complexity, we calculate the Hausdorff fractal dimension [96],
$F_{\textrm{D}}$. As shown in Figs. 5(a) and 5(c), we first numerically
categorize the inter-well and intra-well behavior types. Then, the boundary
between the inter-well and intra-well categories can be extracted. After that,
as illustrated in Figs. 5(b) and 5(d), we implement the box-counting algorithm
[97, 98] to calculate the Hausdorff fractal dimension of extracted boundary
curves in each parameter space plot. We use square-shaped boxes of variable
size $\epsilon$ to cover the boundary curves [e.g., black lines in Figs. 5(b)
and (d)], and count the total number of boxes, $N(\epsilon)$, needed to cover
all boundaries. As $\epsilon\rightarrow 0$, $N(\epsilon)\rightarrow\infty$,
and we obtain $F_{\textrm{D}}$ as,

$F_{\textrm{D}}=\Big{|}\lim_{\epsilon\to
0}\big{(}{\log(N)}/{\log(\frac{1}{\epsilon})}\big{)}\Big{|}.$ (3)

Finally, we use the least-square method to perform linear regression on the
$\log(N)$ vs. $\log(\frac{1}{\epsilon})$ plot to determine $F_{\textrm{D}}$.
Detailed procedures are further explained in the Supplemental Materials [56].
For example, the linear regressions shown in Figs. 5(e) and 5(f) generate
$F_{\textrm{D}}=1.29$ and $F_{\textrm{D}}=1.26$ for the parameter space
boundaries shown in Figs. 5(b) and 5(d), respectively. To verify our
implementation, we also use the software Fraclac/ImageJ [99] developed by the
National Institute of Health (NIH). It determines the fractal dimensions of
Figs. 5(b) and 5(d) as $1.2813$ and $1.2545$, respectively, which closely
match our numerical predictions of $F_{\textrm{D}}$ from Figs. 5(e) and 5(f).
Following the above procedure, we calculate the fractal dimensions of
boundaries in 31 forcing amplitude-frequency plots and 31 damping ratio-
forcing frequency plots of the parameter space (see Supplemental Materials
[56]) for $0.001\leq\gamma\leq 0.30$ and $0.10\leq G\leq 0.25$, respectively.
Using Fraclac/ImageJ [99], we also calculate the average, maximum, and minimum
fractal dimensions in all cases. We summarize all results into Fig. 5(g) for
different damping and Fig. 5(h) for different forcing amplitude, respectively.
Our results indicate that the Hausdorff fractal dimension of the boundaries
between behavior categories decreases with an increase in the damping ratio
and forcing amplitude.
Conclusion: We show that four types of system behaviors (switching, reverting,
vacillating, and intra-well) exist as different regions in the parameter space
of a dissipative bi-stable Duffing oscillator. Our numerical results reveal
the fractal nature of the boundaries between the inter-well and intra-well
regions, and the box-counting algorithm leads to the characterization of the
Hausdorff fractal dimension of these boundaries. This study exclusively
focuses on the fractal patterns in the parameter space, often neglected
compared to fractal patterns in the phase space. These findings may have
important implications for the design of advanced mechanical metamaterials for
shape-morphing applications.

We acknowledge the support from the National Institutes of Health (NIH):
Project No. 1R01EB032959-01. Start-up funds from the Department of Mechanical
Engineering at the Univ. of Utah also supported this work. The support and
resources from the Center for High-Performance Computing at Univ. of Utah are
gratefully acknowledged. The authors thank Prof. Lawrence Virgin (Duke Univ.),
Prof. Suyi Li (Virginia Tech.), and Prof. Andres Arrieta (Purdue Univ.) for
inspirational discussions.

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