Sysout, by definition, outputs to the JVM stdout and stderr I/O channels. Stdout is expected to be a continuous data stream. The only way to route it to rolling files is to first redirect stdout to a file (since the default destination is to the console), then when you want to roll it, stop Tomcat and its JVM, re-assign the stdout channel to a new file, and restart Tomcat. Re-routing of stdout on the fly is not a "write once/run anywhere" capability, so there is no support in pure
Java. Stderr operates identically.
Just for info, the most common Tomcat run scripts redirect the JVM stdout to its logs/catalina.out file. Some people do shut down Tomcat daily and rename this file in order to roll logs.
The secret of how to be miserable is to constantly expect things are going to happen the way that they are "supposed" to happen.
You can have faith, which carries the understanding that you may be disappointed. Then there's being a willfully-blind idiot, which virtually guarantees it.